Theorem necon4abid 2834

Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon4abid.1	|- ( ph -> ( A =/= B <-> -. ps ) )

Assertion

Ref	Expression
necon4abid	|- ( ph -> ( A = B <-> ps ) )

Proof of Theorem necon4abid

Step	Hyp	Ref	Expression
1		notnotb 304	. 2 |- ( ps <-> -. -. ps )
2		necon4abid.1	. . 3 |- ( ph -> ( A =/= B <-> -. ps ) )
3	2	necon1bbid 2833	. 2 |- ( ph -> ( -. -. ps <-> A = B ) )
4	1, 3	syl5rbb 273	1 |- ( ph -> ( A = B <-> ps ) )

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:  necon4bbid  2835  necon2bbid  2837  birthdaylem3  24680  lgsprme0  25064  nmounbi  27631