Theorem necon3bbid 2551

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 192	. . 3 |- (ph -> (A = B <-> ps))
3	2	necon3abid 2550	. 2 |- (ph -> (A =/= B <-> -. ps))
4	3	bicomd 192	1 |- (ph -> (-. ps <-> A =/= B))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642   =/= wne 2517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-ne 2519

This theorem is referenced by:  necon3bid  2552  eldifsn  3840  brltc  6115  addceq0  6220