Proving 2 theorems are equivalent I have 2 theorems of the form:
Theorem 1: if $P_1$, then $Q_1$
Theorem 2: if $P_2$, then $Q_2$
I need to show that the 2 theorems are equivalent. Does that mean that I need to show the following:
Assume Theorem 1 is true and $P_2$ is true, need to show $Q_2$ is true.
Assume Theorem 2 is true and $P_1$ is true, need to show $Q_1$ is true.
Also, if $Q_1$ and $Q_2$ are the same statement, if I need to show Theorem 1 and Theorem 2 are equivalent, is that the same as showing $P_1 \iff P_2$?
 A: Let $T_1 := P_1 \implies Q_1$ and $T_2 := P_2 \implies Q_2$
You are trying to prove
$$T_1 \iff T_2$$
Wikipedia's article on Logical Equivalence offers various strategies for proving $T_1 \iff T_2$.
For example,

$p \iff q \equiv p \implies q \land q \implies p$

which in your case is saying that you can prove
$$T_1 \iff T_2$$
by proving both
$$
\begin{aligned}
T_1 &\implies T_2 &&(1)\\
T_2 &\implies T_1 &&(2)\\
\end{aligned}
$$
So, for example, you would need to prove $(1)$. Either $T_1$ is true or it's false.
If $T_1$ is false, then $(1)$ is vacuously true.
If $T_1$ is true, then you need to prove $T_2$.
So assume $T_1 = P_1 \implies Q_1$ is true.
In $T_2$, if $P_2$ is false, then $T_2$ is vacuously true; if $P_2$ is true, then you need to prove $Q_2$.
So, yes

Assume Theorem $1$ is true and $P_1$ is true, need to show $Q_2$ is true.
Assume Theorem $2$ is true and $P_1$ is true, need to show $Q_2$ is true.

summarize what you need to do prove $(1)$ and $(2)$, respectively.


Also, if $Q_1$ and $Q_2$ are the same statement, if I need to show Theorem $1$ and Theorem $2$ are equivalent, is that the same as showing $P_1 \iff P_2$

Yes.
A: Here's a truth-table answer.

Theorem 1: if $A$, then $R$ Theorem 2: if $B$, then $S$ I need
to show that the 2 theorems are equivalent.


Assume Theorem 1 is true and $B$ is true, need to show $S$ is
true. Assume Theorem 2 is true and $A$ is true, need to show $R$
is true.

Yes, this works:


(if $R$ and $S$ are the same statement) $A \iff B$

This also works; renaming $R$ and $S$ as $Q:$

