A "Rigorous" proof for this seemingly obvious proposition.

In mendelson logic textbook , there is this proposition,(I will replace some letters to make it easier to read)

"If $$P$$ is a tautology containing as statement letters $$A_1,A_2,\ldots,A_n$$ and $$Q$$ arises from $$T$$ by substituting statement forms $$B_1,B_2,\ldots,B_n$$ for $$A_1,A_2,\ldots,A_n,$$ respectively, then $$Q$$ is a tautology;that is, substituting in a tautology yields a tautology"

This proposition seems way too obvious too me. Then I wondered, is there a rigorous proof of this proposition. There was a proof in the textbook:

"Assume that $$P$$ is a tautology. For any assignment of truth values to the statement letters in $$Q$$, the forms $$B_1,B_2,\ldots,B_n$$ have truth values $$x_1,x_2,\ldots,x_n$$ (where each $$x_i$$ is $$T$$ or $$F$$). If we assign the values $$x_1,x_2,...,x_n$$ to $$A_1,A_2,\ldots,A_n,$$ respectively, then the resulting truth value of $$P$$ is the truth value of $$Q$$ for the given assignment of truth values.Since $$P$$ is a tautology, this truth value must be $$T.$$ Thus, $$Q$$ always takes the value $$T$$."

The proposition itself is perfectly understandable and it is a very useful proposition(Maybe even too obvious for me). But the proof seems to be not even slightly understandable to me for some reason. I didn't found any other proofs for this theorem anywhere. Is there a better proof for this theorem or I am missing something? (Maybe the proposition is so obvious that proving it is impossible?)

Edit:I just noticed that when I said "I will replace some letters to make it easier to read" then that means "this proposition will stay true nomatter if i replace some symbols". But the proposition itself is "this proposition will stay true no matter if i replace some symbols". So it is like a propositon inside itself.So in shorter words, I am applying this proposition without even proving it in the first place. This is by far the weirdest part about this proposition to me.

• The proof in the textbook is rigorous. Nov 26, 2020 at 8:13
• A tautology is ALWAYS true, i.e. every line of the truth table outputs the TRUE value. IF we replace e.g. $p$ with formula $A$ whatever, what truth values $A$ can have? Either TRUE or FALSE; but in both cases the corresponding lines of the truth table we have true as output. Nov 26, 2020 at 8:15
• I agree with you, and on occasion I've seen things like this myself, namely where being able to intuitively grasp the correctness of a proof seems to require essentially the same "metalogic-comprehension awareness proficiency" that one would need to in order to initially intuitively grasp the correctness of the result itself. The result itself, of course, does say something meaningful about the formal language, since you can use the result itself to deduce the tautology'ness of some wff's by immediately recognizing them as "instances" of known tautologies. Nov 26, 2020 at 8:59
• What should I do if I find myself in this kind of weird situation when trying to prove a proposition "rigorously" ? Thanks for your thoughts about this problem. Nov 26, 2020 at 9:48
• One of the difficulties I used to have (and still do to some extent) is that, when working at this "level of intuitive obviousness", intermediate level logic texts (Mendelson level) bring ideas of functions and variable substitution in functions onto the scene supposedly (or at least many readers will think so) to provide more rigor. However, using functions and such to provide these notions of unique output'ness and meta-linguistic variable usage seem to me, if anything, further removed from "intuitive obviousness" than the things they are trotted out to provide rigor for. (continued) Nov 26, 2020 at 12:17

Long comment

Maybe, a more lengthy approach can help, considering the case of a single atom $$p$$.

Let $$F$$ a formula and let $$p$$ a propositional symbol occurring in $$F$$. This means that the formula is a truth-function $$F(p)$$: for every truth value assigned to $$p$$, the truth table corresponding to $$F$$ will outputs a truth value.

But $$F$$ is a tautology: thus for every assignments of truth values to atoms, the truth table will produce the value TRUE.

Consider now the formula $$F' := F[A/p]$$, where $$A$$ is a formula whatever. Irrespective of the "form" of $$A$$, every assignment will output either TRUE or FALSE as truth value for the formula.

Thus, when $$A$$ is evaluated to TRUE, we have to consider the lines in the original truth table for $$F$$ where $$p$$ is evaluated to TRUE. Due to the fact that $$A$$ is a tautology, in those lines the formula $$F$$ has value TRUE; thus also $$F'$$ will have TRUE in those lines.

But also when in the original truth table for $$F$$ the atom $$p$$ is evaluated to FALSE, the corresponding lines for formula $$F$$ have value TRUE; thus also $$F'$$ will have TRUE in those lines.

The argument can be iterated, taking into account that a formula is an expression of finite length and thus only a finite number of atoms occurs into it.