Theorem necon3ai 2463

Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

Hypothesis

Ref	Expression
necon3ai.1	|- ( ph -> A = B )

Assertion

Ref	Expression
necon3ai	|- ( A =/= B -> -. ph )

Proof of Theorem necon3ai

Step	Hyp	Ref	Expression
1		df-ne 2415	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ai.1	. . 3 |- ( ph -> A = B )
3	2	con3i 637	. 2 |- ( -. A = B -> -. ph )
4	1, 3	sylbi 121	1 |- ( A =/= B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   =/= wne 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2415

This theorem is referenced by:  nelsn  3729  disjsn2  3757  0nelxp  4782  fvunsng  5883  map0b  6934  difinfsnlem  7403  hashprg  11198  gcd1  12708  gcdzeq  12743  phimullem  12947  pcgcd1  13051  pc2dvds  13053  pockthlem  13079  znrrg  14934  mpodvdsmulf1o  15984  2sqlem8  16122