Consequence in Logic For arbitrary formulas $A,B,C$ it holds that: 1. $\\{A,B\\} \vDash C $ if $A \vDash (B \Rightarrow C)$ 2. $(A \Rightarrow B) \vDash C$ if $A \vDash (B \Rightarrow C)$ 3. $A \vDash C$ if $A \vDash (B \Rightarrow C)$ I know that only first one holds, can someone explain me why?

Here's one approach:

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

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