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Theorem necon3bi 2305
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3bi.1  |-  ( A  =  B  ->  ph )
Assertion
Ref Expression
necon3bi  |-  ( -. 
ph  ->  A  =/=  B
)

Proof of Theorem necon3bi
StepHypRef Expression
1 necon3bi.1 . . 3  |-  ( A  =  B  ->  ph )
21con3i 597 . 2  |-  ( -. 
ph  ->  -.  A  =  B )
3 df-ne 2256 . 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 1289    =/= wne 2255
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 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  pwne  3987  nltpnft  9248  ngtmnft  9249
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