Theorem necon3ai 2451

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 2403	. 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 1397   =/= wne 2402

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 2403

This theorem is referenced by:  nelsn  3704  disjsn2  3732  0nelxp  4753  fvunsng  5847  map0b  6855  difinfsnlem  7297  hashprg  11071  gcd1  12557  gcdzeq  12592  phimullem  12796  pcgcd1  12900  pc2dvds  12902  pockthlem  12928  znrrg  14673  mpodvdsmulf1o  15713  2sqlem8  15851