Theorem impbidd 181

Description: Deduce an equivalence from two implications. (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		bi3 179	. 2 |- ((ch -> th) -> ((th -> ch) -> (ch <-> th)))
4	1, 2, 3	syl6c 60	1 |- (ph -> (ps -> (ch <-> th)))

Syntax hints:   -> wi 4   <-> wb 176

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 177

This theorem is referenced by:  impbid21d  182  pm5.74  235