Theorem necon3abid 2244

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 2206	. 2 |- ( A =/= B <-> -. A = B )
2		necon3abid.1	. . 3 |- ( ph -> ( A = B <-> ps ) )
3	2	notbid 592	. 2 |- ( ph -> ( -. A = B <-> -. ps ) )
4	1, 3	syl5bb 181	1 |- ( ph -> ( A =/= B <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   = wceq 1243   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-ne 2206

This theorem is referenced by:  necon3bbid  2245  fndmdif  5272  expnegap0  9263