Theorem necon1bi 2560

Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon1bi.1	|- (A =/= B -> ph)

Assertion

Ref	Expression
necon1bi	|- (-. ph -> A = B)

Proof of Theorem necon1bi

Step	Hyp	Ref	Expression
1		necon1bi.1	. . 3 |- (A =/= B -> ph)
2	1	con3i 127	. 2 |- (-. ph -> -. A =/= B)
3		nne 2521	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	sylib 188	1 |- (-. ph -> A = B)

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:  mapprc  6005