TITLE: Question about implication
QUESTION [9 upvotes]: I have question about something i don't understand.
$\alpha$, $\beta$, and $\gamma$ are statements.
if $\alpha\implies\beta\lor\gamma$ then it's necessary that $\alpha\implies\beta$ or $\alpha\implies\gamma$. that answer is "no" but i can't understand why.
if it's given that $\alpha\implies\beta\lor\gamma$ then it means that when $\alpha$ is true then $\beta\lor\gamma$ must be also true which means that either $\beta$ is true or $\gamma$ is true. so it means that $\alpha\implies\beta$ or $\alpha\implies\gamma$, isn't it? because the only case when it's might be not true is when $\alpha$ is true, $\beta$ is false and also $\gamma$ is false.
So i can't understand why the answer to this question is "no".
I hope someone can explain it.
Thank you. (P.S. sorry for my bad English).

REPLY [3 votes]: My summary of the two interpretations of your question:
Is $(a \Rightarrow (b \vee c)) \Rightarrow ((a \Rightarrow b) \vee (a \Rightarrow c))$ a tautology?
Yes.
If $a \vdash b \vee c$ then does $ a \vdash b$ or  $a \vdash c$?
No.