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Theorem necon4ai 2665
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
necon4ai.1  |-  ( A  =/=  B  ->  -.  ph )
Assertion
Ref Expression
necon4ai  |-  ( ph  ->  A  =  B )

Proof of Theorem necon4ai
StepHypRef Expression
1 necon4ai.1 . . 3  |-  ( A  =/=  B  ->  -.  ph )
21con2i 115 . 2  |-  ( ph  ->  -.  A  =/=  B
)
3 nne 2607 . 2  |-  ( -.  A  =/=  B  <->  A  =  B )
42, 3sylib 190 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    =/= wne 2601
This theorem is referenced by:  dmsn0el  5341  cfeq0  8138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 179  df-ne 2603
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