Consequence in Logic 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?
 A: The canonical way to show that the two last implications don't hold would be to find formulas you can plug in for $A$, $B$ and $C$, such that the entailment to the right of the "if" is logically valid, but the one to the left isn't.
For example, try setting $B\equiv P$, $C\equiv Q$ and $A\equiv (P\Rightarrow Q)$, where $P$ and $Q$ are propositional variables.
A: 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 $A \vDash C$.
A: Number 3 Should be fairly obvious:
A entails (B implies C) 

Which, in English, becomes:
A entails C or A entails not B

Number 2 is a bit more complex. The values it fails for are $A, ¬B, ¬C$.
