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Theorem necon1abii 2710
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Hypothesis
Ref Expression
necon1abii.1  |-  ( -. 
ph 
<->  A  =  B )
Assertion
Ref Expression
necon1abii  |-  ( A  =/=  B  <->  ph )

Proof of Theorem necon1abii
StepHypRef Expression
1 notnot 291 . 2  |-  ( ph  <->  -. 
-.  ph )
2 necon1abii.1 . . 3  |-  ( -. 
ph 
<->  A  =  B )
32necon3bbii 2709 . 2  |-  ( -. 
-.  ph  <->  A  =/=  B
)
41, 3bitr2i 250 1  |-  ( A  =/=  B  <->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1370    =/= wne 2644
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 2646
This theorem is referenced by:  necon2abii  2714  marypha1lem  7787  npomex  9269  restutopopn  19938
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