Theorem imbi12

Description: Closed form of imbi12i . Was automatically derived from its "Virtual Deduction" version and the Metamath program "MM-PA> MINIMIZE__WITH *" command. (Contributed by Alan Sare, 18-Mar-2012)

	Ref	Expression
Assertion	imbi12	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\chi \leftrightarrow \theta \right)\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		simplim	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\to \neg \left(\chi \leftrightarrow \theta \right)\right)\to \left(\phi \leftrightarrow \psi \right)$$
2		simprim	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\to \neg \left(\chi \leftrightarrow \theta \right)\right)\to \left(\chi \leftrightarrow \theta \right)$$
3	1 2	imbi12d	$$⊢\neg \left(\left(\phi \leftrightarrow \psi \right)\to \neg \left(\chi \leftrightarrow \theta \right)\right)\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)$$
4	3	expi	$$⊢\left(\phi \leftrightarrow \psi \right)\to \left(\left(\chi \leftrightarrow \theta \right)\to \left(\left(\phi \to \chi \right)\leftrightarrow \left(\psi \to \theta \right)\right)\right)$$