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 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 623  ax-in2 624

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

This theorem is used by:  nelsn  3744  disjsn2  3772  0nelxp  4804  fvunsng  5912  map0b  6972  difinfsnlem  7445  hashprg  11271  gcd1  12789  gcdzeq  12824  phimullem  13032  pcgcd1  13136  pc2dvds  13138  pockthlem  13164  znrrg  15104  mpodvdsmulf1o  16284  ppiqub  16293  2sqlem8  16447