Theorem necon3ai 2413

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 2365	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ai.1	. . 3 |- ( ph -> A = B )
3	2	con3i 633	. 2 |- ( -. A = B -> -. ph )
4	1, 3	sylbi 121	1 |- ( A =/= B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   =/= wne 2364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2365

This theorem is referenced by:  nelsn  3653  disjsn2  3681  0nelxp  4687  fvunsng  5752  map0b  6741  difinfsnlem  7158  hashprg  10879  gcd1  12124  gcdzeq  12159  phimullem  12363  pcgcd1  12466  pc2dvds  12468  pockthlem  12494  znrrg  14148  2sqlem8  15210