Theorem impbid21d 128

Description: Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)

Hypotheses

Ref	Expression
impbid21d.1	|- ( ps -> ( ch -> th ) )
impbid21d.2	|- ( ph -> ( th -> ch ) )

Assertion

Ref	Expression
impbid21d	|- ( ph -> ( ps -> ( ch <-> th ) ) )

Proof of Theorem impbid21d

Step	Hyp	Ref	Expression
1		impbid21d.1	. . 3 |- ( ps -> ( ch -> th ) )
2	1	a1i 9	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
3		impbid21d.2	. . 3 |- ( ph -> ( th -> ch ) )
4	3	a1d 22	. 2 |- ( ph -> ( ps -> ( th -> ch ) ) )
5	2, 4	impbidd 127	1 |- ( ph -> ( ps -> ( ch <-> th ) ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  impbid  129  pm5.1im  173