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Theorem necon2ai 2362
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2ai.1  |-  ( A  =  B  ->  -.  ph )
Assertion
Ref Expression
necon2ai  |-  ( ph  ->  A  =/=  B )

Proof of Theorem necon2ai
StepHypRef Expression
1 necon2ai.1 . . 3  |-  ( A  =  B  ->  -.  ph )
21con2i 616 . 2  |-  ( ph  ->  -.  A  =  B )
3 df-ne 2309 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3sylibr 133 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    =/= wne 2308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2309
This theorem is referenced by:  necon2i  2364  neneqad  2387  intexr  4075  iin0r  4093  tfrlemisucaccv  6222  pm54.43  7046  renepnf  7820  renemnf  7821  lt0ne0d  8282  nnne0  8755  nn0nepnf  9055  hashennn  10533  bj-intexr  13120
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