Theorem necon1abid 2832

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)

Hypothesis

Ref	Expression
necon1abid.1	|- ( ph -> ( -. ps <-> A = B ) )

Assertion

Ref	Expression
necon1abid	|- ( ph -> ( A =/= B <-> ps ) )

Proof of Theorem necon1abid

Step	Hyp	Ref	Expression
1		notnotb 304	. 2 |- ( ps <-> -. -. ps )
2		necon1abid.1	. . 3 |- ( ph -> ( -. ps <-> A = B ) )
3	2	necon3bbid 2831	. 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:  lttri2  10120  xrlttri2  11975  ioon0  12201  lssne0  18951  xmetgt0  22163  sotrine  31658