Theorem necon3ad 2037

Description: Contrapositive law deduction for inequality. (The proof was 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 2021	. . 3 |- (-. A =/= B <-> A = B)
3	1, 2	syl6ibr 230	. 2 |- (ph -> (ps -> -. A =/= B))
4	3	con2d 107	1 |- (ph -> (A =/= B -> -. ps))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   =/= wne 2017

This theorem is referenced by:  necon3d 2041  disjpss 2924  nlt1pi 6185  0nnei 9002  ocnel 10803  hatomistici 11934  dmse1 15001  pltnle 16786  atnlt 17009  atomnle 17016  hlatmstc 17047

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 164  df-ne 2019

Copyright terms: Public domain