Theorem biimpexp

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

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

Proof

Step	Hyp	Ref	Expression
1		dfbi2	|- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	imbi1i	|- ( ( ( ph <-> ps ) -> ch ) <-> ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ch ) )
3		impexp	|- ( ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ch ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) )
4	2 3	bitri	|- ( ( ( ph <-> ps ) -> ch ) <-> ( ( ph -> ps ) -> ( ( ps -> ph ) -> ch ) ) )