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Theorem necon1abii 2657
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)
Hypothesis
Ref Expression
necon1abii.1  |-  ( -. 
ph 
<->  A  =  B )
Assertion
Ref Expression
necon1abii  |-  ( A  =/=  B  <->  ph )

Proof of Theorem necon1abii
StepHypRef Expression
1 df-ne 2603 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon1abii.1 . . 3  |-  ( -. 
ph 
<->  A  =  B )
32con1bii 331 . 2  |-  ( -.  A  =  B  <->  ph )
41, 3bitri 249 1  |-  ( A  =/=  B  <->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1369    =/= wne 2601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-ne 2603
This theorem is referenced by:  necon2abii  2661  marypha1lem  7675  npomex  9157  restutopopn  19788
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