1. ## Question about verifying equivalence

Can someone please explain how to verify/prove that:
((P => R1) ^ (P => R2)) <=> (P => (R1 ^ R2))
and
((R1 => P) ^ (R2 => P)) <=> ((R1 v R2) => P)
where "^" means "and", "=>" means "implies, and "<=>" means "equivalent to".
They're obviously equivalent from glance, but I just don't know how to mathematically verify them, ie, manipulate left and right sides so that they're identical.

Thanks.

2. Recall that $\displaystyle \left( {A \Rightarrow B} \right) \equiv \left( {\neg A \vee B} \right)$.
So $\displaystyle \left( {R_1 \Rightarrow P} \right) \wedge \left( {R_2 \Rightarrow P} \right) \equiv \left( {\neg R_1 \vee P} \right) \wedge \left( {\neg R_2 \vee P} \right) \equiv \left( {\neg R_1 \wedge \neg R_2 } \right) \vee P$
But $\displaystyle \left( {\neg R_1 \wedge \neg R_2 } \right) \vee P \equiv \neg \left( {R_1 \vee R_2 } \right) \vee P \equiv \left( {R_1 \vee R_2 } \right) \Rightarrow P$.

Now you show the others.

3. Thank you!
Worked perfectly (=