Theorem necon3abid 2403

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 2365	. 2 |- ( A =/= B <-> -. A = B )
2		necon3abid.1	. . 3 |- ( ph -> ( A = B <-> ps ) )
3	2	notbid 668	. 2 |- ( ph -> ( -. A = B <-> -. ps ) )
4	1, 3	bitrid 192	1 |- ( ph -> ( A =/= B <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   =/= wne 2364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2365

This theorem is referenced by:  necon3bbid  2404  fndmdif  5663  expnegap0  10618  gcdn0gt0  12115  cncongr2  12242  mulgnegnn  13202  domnmuln0  13769