Theorem necon4bbid 2835

Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)

Hypothesis

Ref	Expression
necon4bbid.1	|- ( ph -> ( -. ps <-> A =/= B ) )

Assertion

Ref	Expression
necon4bbid	|- ( ph -> ( ps <-> A = B ) )

Proof of Theorem necon4bbid

Step	Hyp	Ref	Expression
1		necon4bbid.1	. . . 4 |- ( ph -> ( -. ps <-> A =/= B ) )
2	1	bicomd 213	. . 3 |- ( ph -> ( A =/= B <-> -. ps ) )
3	2	necon4abid 2834	. 2 |- ( ph -> ( A = B <-> ps ) )
4	3	bicomd 213	1 |- ( ph -> ( ps <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   =/= wne 2794

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  df-ne 2795

This theorem is referenced by:  fzn  12357  lgsqr  25076