TITLE: Consequence in Logic
QUESTION [5 upvotes]: For arbitrary formulas $A,B,C$ it holds that:

$\{A,B\} \vDash C $ if $A \vDash (B \Rightarrow C)$
$(A \Rightarrow B) \vDash C$ if $A \vDash (B \Rightarrow C)$
$A \vDash C$ if $A \vDash (B \Rightarrow C)$

I know that only first one holds, can someone explain me why?

REPLY [2 votes]: Here's one approach:

Note that trivially $\vDash p \to p$, so a fortiori $p \vDash p \to p$. But  $p \to p \nvDash p$ (suppose $p$ is false). So we can have an instance  of $A \vDash B \to C$ without the corresponding $A \to B \vDash C$.
Note that trivially $\vDash q \to q$, so a fortiori $p \vDash q \to q$. But of course $p \nvDash q$. So we can have an instance of $A \vDash (B \Rightarrow C)$ without the corresponding $A \vDash C$.