Theorem necon4bd 2579

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon4bd.1	|- (ph -> (-. ps -> A =/= B))

Assertion

Ref	Expression
necon4bd	|- (ph -> (A = B -> ps))

Proof of Theorem necon4bd

Step	Hyp	Ref	Expression
1		nne 2521	. 2 |- (-. A =/= B <-> A = B)
2		necon4bd.1	. . 3 |- (ph -> (-. ps -> A =/= B))
3	2	con1d 116	. 2 |- (ph -> (-. A =/= B -> ps))
4	1, 3	syl5bir 209	1 |- (ph -> (A = B -> ps))

Syntax hints:  -. wn 3   -> wi 4   = 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)