Theorem impbid21d 182

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 10	. 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 181	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:  impbid  183  pm5.1im  229