Theorem necon2ad 2564

Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon2ad.1	|- (ph -> (A = B -> -. ps))

Assertion

Ref	Expression
necon2ad	|- (ph -> (ps -> A =/= B))

Proof of Theorem necon2ad

Step	Hyp	Ref	Expression
1		nne 2520	. . 3 |- (-. A =/= B <-> A = B)
2		necon2ad.1	. . 3 |- (ph -> (A = B -> -. ps))
3	1, 2	syl5bi 208	. 2 |- (ph -> (-. A =/= B -> -. ps))
4	3	con4d 97	1 |- (ph -> (ps -> A =/= B))

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2516

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 2518

This theorem is referenced by:  necon2d  2566  nulnnn  4556  enadjlem1  6059