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Theorem necon2ai 2300
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 590 . 2  |-  ( ph  ->  -.  A  =  B )
3 df-ne 2247 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
42, 3sylibr 132 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    =/= wne 2246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2247
This theorem is referenced by:  necon2i  2302  neneqad  2325  intexr  3933  iin0r  3951  tfrlemisucaccv  5974  pm54.43  6518  renepnf  7228  renemnf  7229  lt0ne0d  7681  nnne0  8134  nn0nepnf  8426  sizeennn  9804  bj-intexr  10857
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