Theorem biimpexp 31597

Description: A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)

Assertion

Ref	Expression
biimpexp	|- ( ( ( ph <-> ps ) -> ch ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) )

Proof of Theorem biimpexp

Step	Hyp	Ref	Expression
1		dfbi2 660	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	imbi1i 339	. 2 |- ( ( ( ph <-> ps ) -> ch ) <-> ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ch ) )
3		impexp 462	. 2 |- ( ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ch ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) )
4	2, 3	bitri 264	1 |- ( ( ( ph <-> ps ) -> ch ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  axextdfeq  31703