Theorem necon2ai 2561

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

Hypothesis

Ref	Expression
necon2ai.1	|- (A = B -> -. ph)

Assertion

Ref	Expression
necon2ai	|- (ph -> A =/= B)

Proof of Theorem necon2ai

Step	Hyp	Ref	Expression
1		nne 2520	. . 3 |- (-. A =/= B <-> A = B)
2		necon2ai.1	. . 3 |- (A = B -> -. ph)
3	1, 2	sylbi 187	. 2 |- (-. A =/= B -> -. ph)
4	3	con4i 122	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:  necon2i  2563  neneqad  2586  peano4  4557  addccan2  4559  ncssfin  6151  ncspw1eu  6159  nntccl  6170  ceclb  6183  ce0ncpw1  6185  cecl  6186  nclecid  6197  le0nc  6200  addlecncs  6209