Theorem necon3abid 2285

Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.)

Hypothesis

Ref	Expression
necon3abid.1	|- ( ph -> ( A = B <-> ps ) )

Assertion

Ref	Expression
necon3abid	|- ( ph -> ( A =/= B <-> -. ps ) )

Proof of Theorem necon3abid

Step	Hyp	Ref	Expression
1		df-ne 2247	. 2 |- ( A =/= B <-> -. A = B )
2		necon3abid.1	. . 3 |- ( ph -> ( A = B <-> ps ) )
3	2	notbid 625	. 2 |- ( ph -> ( -. A = B <-> -. ps ) )
4	1, 3	syl5bb 190	1 |- ( ph -> ( A =/= B <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   = wceq 1285   =/= wne 2246

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578

This theorem depends on definitions:  df-bi 115  df-ne 2247

This theorem is referenced by:  necon3bbid  2286  fndmdif  5304  expnegap0  9581  gcdn0gt0  10513  cncongr2  10630