Theorem necon1abid 2570

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.)

Hypothesis

Ref	Expression
necon1abid.1	|- (ph -> (-. ps <-> A = B))

Assertion

Ref	Expression
necon1abid	|- (ph -> (A =/= B <-> ps))

Proof of Theorem necon1abid

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 |- (A =/= B <-> -. A = B)
2		necon1abid.1	. . 3 |- (ph -> (-. ps <-> A = B))
3	2	con1bid 320	. 2 |- (ph -> (-. A = B <-> ps))
4	1, 3	syl5bb 248	1 |- (ph -> (A =/= B <-> ps))

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: (None)