Theorem impbidd 200

Description: Deduce an equivalence from two implications. Double deduction associated with impbi 198 and impbii 199. Deduction associated with impbid 202. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Hypotheses

Ref	Expression
impbidd.1	|- ( ph -> ( ps -> ( ch -> th ) ) )
impbidd.2	|- ( ph -> ( ps -> ( th -> ch ) ) )

Assertion

Ref	Expression
impbidd	|- ( ph -> ( ps -> ( ch <-> th ) ) )

Proof of Theorem impbidd

Step	Hyp	Ref	Expression
1		impbidd.1	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
2		impbidd.2	. 2 |- ( ph -> ( ps -> ( th -> ch ) ) )
3		impbi 198	. 2 |- ( ( ch -> th ) -> ( ( th -> ch ) -> ( ch <-> th ) ) )
4	1, 2, 3	syl6c 70	1 |- ( ph -> ( ps -> ( ch <-> th ) ) )

Syntax hints:   -> wi 4   <-> wb 196

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

This theorem is referenced by:  impbid21d  201  pm5.74  259  seglecgr12  32218  prtlem18  34162