Theorem necon4ai 2575

Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon4ai.1	|- (A =/= B -> -. ph)

Assertion

Ref	Expression
necon4ai	|- (ph -> A = B)

Proof of Theorem necon4ai

Step	Hyp	Ref	Expression
1		necon4ai.1	. . 3 |- (A =/= B -> -. ph)
2	1	con2i 112	. 2 |- (ph -> -. A =/= B)
3		nne 2520	. 2 |- (-. A =/= B <-> A = B)
4	2, 3	sylib 188	1 |- (ph -> A = B)

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2516

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 2518

This theorem is referenced by: (None)