Theorem necon3bbid 2407

Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.)

Hypothesis

Ref	Expression
necon3bbid.1	|- ( ph -> ( ps <-> A = B ) )

Assertion

Ref	Expression
necon3bbid	|- ( ph -> ( -. ps <-> A =/= B ) )

Proof of Theorem necon3bbid

Step	Hyp	Ref	Expression
1		necon3bbid.1	. . . 4 |- ( ph -> ( ps <-> A = B ) )
2	1	bicomd 141	. . 3 |- ( ph -> ( A = B <-> ps ) )
3	2	necon3abid 2406	. 2 |- ( ph -> ( A =/= B <-> -. ps ) )
4	3	bicomd 141	1 |- ( ph -> ( -. ps <-> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   =/= wne 2367

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 2368

This theorem is referenced by:  necon3bid  2408  eldifsn  3749  prmrp  12313  4sqlem17  12576  nzrunit  13744  lgsne0  15279  2sqlem7  15362