Theorem necon3abid 2345

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 2307	. 2 |- ( A =/= B <-> -. A = B )
2		necon3abid.1	. . 3 |- ( ph -> ( A = B <-> ps ) )
3	2	notbid 656	. 2 |- ( ph -> ( -. A = B <-> -. ps ) )
4	1, 3	syl5bb 191	1 |- ( ph -> ( A =/= B <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1331   =/= wne 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2307

This theorem is referenced by:  necon3bbid  2346  fndmdif  5518  expnegap0  10294  gcdn0gt0  11655  cncongr2  11774