How to prove syntactic consequence? I have a problem like this:
$$ F = \{A\sim B, B \sim C, ¬C∨B\} $$
And I want to check if this is true: $\{A∨C\}\vdash F$.
I will assume for the purposes of this answer that $A \sim B$ means $\text{$A$ if and only if $B$}$.
I'm also going to assume that you want to show that the following is true when interpreted as a sequent.
$$ A \lor C \vdash A \leftrightarrow B, B \leftrightarrow C, \lnot C \lor B $$
Let's rewrite this slightly
$$ A \lor C, C \vdash A \leftrightarrow B, B \leftrightarrow C, B $$
This statement is not a tautology because if $A$ and $C$ are true, but $B$ is false, then all the premises are true, but the conclusion is false (in fact, all the disjuncts are false).
I will assume for the purposes of this answer that $A \sim B$ means $\text{$A$ if and only if $B$}$.
I'm also going to assume that you want to show that the following is true when interpreted as a sequent.
$$ A \lor C \vdash A \leftrightarrow B, B \leftrightarrow C, \lnot C \lor B $$
Let's rewrite this slightly
$$ A \lor C, C \vdash A \leftrightarrow B, B \leftrightarrow C, B $$
This statement is not a tautology because if $A$ and $C$ are true, but $B$ is false, then all the premises are true, but the conclusion is false (in fact, all the disjuncts are false).