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Theorem necon2bi 2301
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
Hypothesis
Ref Expression
necon2bi.1  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
necon2bi  |-  ( A  =  B  ->  -.  ph )

Proof of Theorem necon2bi
StepHypRef Expression
1 necon2bi.1 . . 3  |-  ( ph  ->  A  =/=  B )
21neneqd 2267 . 2  |-  ( ph  ->  -.  A  =  B )
32con2i 590 1  |-  ( A  =  B  ->  -.  ph )
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-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-ne 2247
This theorem is referenced by:  minel  3312  rzal  3346  difsnb  3536  fin0  6419  0npi  6565  0nsr  6988  renfdisj  7239  nltpnft  8960  ngtmnft  8961  xrrebnd  8962  sizenncl  9820  rennim  10026
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