Theorem necon3ad 2552

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3ad.1	|- (ph -> (ps -> A = B))

Assertion

Ref	Expression
necon3ad	|- (ph -> (A =/= B -> -. ps))

Proof of Theorem necon3ad

Step	Hyp	Ref	Expression
1		necon3ad.1	. . 3 |- (ph -> (ps -> A = B))
2		nne 2520	. . 3 |- (-. A =/= B <-> A = B)
3	1, 2	syl6ibr 218	. 2 |- (ph -> (ps -> -. A =/= B))
4	3	con2d 107	1 |- (ph -> (A =/= B -> -. ps))

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:  necon3d  2554  disjpss  3601  sfinltfin  4535  nchoicelem8  6296