Theorem necon1bd 2585

Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1bd.1	|- (ph -> (A =/= B -> ps))

Assertion

Ref	Expression
necon1bd	|- (ph -> (-. ps -> A = B))

Proof of Theorem necon1bd

Step	Hyp	Ref	Expression
1		necon1bd.1	. . 3 |- (ph -> (A =/= B -> ps))
2	1	con3d 125	. 2 |- (ph -> (-. ps -> -. A =/= B))
3		nne 2521	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	syl6ib 217	1 |- (ph -> (-. ps -> A = B))

This proof depends on syntax axioms:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  fvclss  5463