Theorem necon3ai 2469

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 2421	. 2 |- ( A =/= B <-> -. A = B )
2		necon3ai.1	. . 3 |- ( ph -> A = B )
3	2	con3i 641	. 2 |- ( -. A = B -> -. ph )
4	1, 3	sylbi 121	1 |- ( A =/= B -> -. ph )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

This theorem depends on definitions:  df-bi 117  df-ne 2421

This theorem is referenced by:  nelsn  3740  disjsn2  3768  0nelxp  4797  fvunsng  5900  map0b  6958  difinfsnlem  7429  hashprg  11227  gcd1  12742  gcdzeq  12777  phimullem  12981  pcgcd1  13085  pc2dvds  13087  pockthlem  13113  znrrg  14967  mpodvdsmulf1o  16018  2sqlem8  16156