# Can we replace propositional variables in the proof for a tautology with variables that stand for formulas/True/False?

For instance we know that $\left(p\wedge q\right)\rightarrow p$ is a tautology and the proof for it is: $$\left(\left(p\wedge q\right)\rightarrow p\right)\equiv\left(\neg\left(p\wedge q\right)\vee p\right)$$ $$\equiv\left(\left(\neg p\vee \neg q\right)\vee p\right)$$ $$\equiv\left(\left(\neg p\vee p\right)\vee\neg q\right)$$ $$\equiv\left(True\vee\neg q\right)$$ $$\equiv True$$ Now if we could replace p and q in the proof with A and B which are variables standing for formulas/True/False to get: $$\left(\left(A\wedge B\right)\rightarrow A\right)\equiv\left(\neg\left(A\wedge B\right)\vee A\right)$$ $$\equiv\left(\left(\neg A\vee \neg B\right)\vee A\right)$$ $$\equiv\left(\left(\neg A\vee A\right)\vee\neg B\right)$$ $$\equiv\left(True\vee\neg B\right)$$ $$\equiv True$$ Then every formula which is achieved by replacing A and B in $\left(\left(A\wedge B\right)\rightarrow A\right)$ with formulas/True/False is proved, like $\left(p\wedge \left(q\vee r\right)\right)\rightarrow p$, $\left(\left(\neg p\vee r\right)\wedge q\right)\rightarrow \left(\neg p\vee r\right)$ and $\left(p\wedge False\right)\rightarrow p$. So I wanna know if there's anything wrong with that replacement

• there is no problem. it is ok. – OmG Aug 30 '17 at 6:31
• @OmG that answering speed thou :) – Pooria Aug 30 '17 at 6:32
• You have already asked it here. – Mauro ALLEGRANZA Aug 30 '17 at 7:10
• I think they're different questions, and I couldn't understand your answer, but I think your comment mentioned what I'm doing here – Pooria Aug 30 '17 at 7:37

It does not hold for properties and relations that involve mere 'possibility'. For example, $P$ is a logical contingency, but if you substitute something for $P$ the result may no longer be a contingency (e.g substitute $A \land \neg A$, and the result is not a contingency). Likewise, it does not work for logical consistency: $\{ P, Q \}$ is logically consistent, but substitute $A$ for $P$, and $\neg A$ for $Q$, and it is no longer logically consistent.