Replacement of sentence symbols by $0$-place connectives Suppose we add $0$-place connectives $\top, \bot$ to our language. For each well-formed formula wff $, \phi$ and sentence symbol, $A,$ let $\phi^{A}_{\top}$ be the wff obtained from $\phi$ by replacing $A$ with $\top.$ Similarly for $\phi^{A}_{\bot}.$ Let $\phi^{A}_{*}= (\phi^{A}_{\top} \lor \phi^{A}_{\bot}).$ 
(i) Suppose $\phi \models\psi$ and $A$ does not appear in $\psi,$ could anyone advise me on how to show $\phi^{A}_{*} \models \psi$ and (ii )how to show $\phi$ is satisfiable iff $\phi^{A}_{*}$ is satisfiable? 
Hints will suffice. Thank you. 
 A: You must start with the definition of tautological implication (we may use this because we are considering only sentence letters) :

$\phi \models\psi$ iff, for every valuation $v$ such that $|\phi|_v = t$, also $|\psi|_v = t$.

Now assume that $\phi$ is built from $n$ sentence letters : $A, A_1, ... A_{n-1}$.
Consider now the truth-table (with $2^n$ rows) for $\phi$ : in these table, half of the rows have $A$ evaluated at t and half evaluated at f.
When we build the formula $\phi^{A}_{\top}$, this formula is equivalent to all the rows of the t-t for $\phi$ where $A$ is evaluated to t; the same with $\phi^{A}_{\bot}$, that is equivalent to the rows with $A$ evaluated to f.
So the formula $\phi^{A}_{*} = (\phi^{A}_{\top} \lor \phi^{A}_{\bot})$ may replace all the truth-table for $\phi$.
Now, go back to the tautological impliction :

$\phi \models\psi$ iff for every valuation $v$ such that $|\phi|_v = t$, also $|\psi|_v = t$,

i.e. for every row in the t-t for $\phi$ that evaluates to t, also  $\psi$ is evaluated to f. 
But the rows of the t-t for $\phi$ which evaluate to t may have $A=t$ or $A=f$; both cases are covered by the disjunction $\phi^{A}_{\top} \lor \phi^{A}_{\bot}$ (if the row has $A=t$, this row is "covered" by $\phi^{A}_{\top}$ and if the row has $A=f$, it is covered by$\phi^{A}_{\bot}$.
So the taut implication still hold : of course, $A$ must not be in $\psi$, because substituting $A$ with $\top$ or $\bot$, we have lost "the link" between the two original formuals when we use the valuation $v$. 
Now I think that you can easily use this argument to show satisfiability...
A: I'm assuming that we're talking first-order predicate logic here and that $\models$ refers to logical consequence. You could first show that $\phi_* ^{A} \models \phi$. Because $\models$ is transitive, the result you're looking for follows from that.
