Theorem necon2ad 1612

Description: Contrapositive inference for inequality.

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		necon2ad.1	. . 3 |- (ph -> (A = B -> -. ps))
2	1	con2d 91	. 2 |- (ph -> (ps -> -. A = B))
3		df-ne 1585	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl6ibr 213	1 |- (ph -> (ps -> A =/= B))

Syntax hints:  -. wn 2   -> wi 3   = wceq 955   =/= wne 1583

This theorem is referenced by:  tz7.2 2927  nordeq 2963  normgt0t 8949  nmcopexlem1 9907  nmcfnexlem1 9936

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-ne 1585

Copyright terms: Public domain