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Theorem necon3ad 2350
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ad.1  |-  ( ph  ->  ( ps  ->  A  =  B ) )
Assertion
Ref Expression
necon3ad  |-  ( ph  ->  ( A  =/=  B  ->  -.  ps ) )

Proof of Theorem necon3ad
StepHypRef Expression
1 df-ne 2309 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
2 necon3ad.1 . . 3  |-  ( ph  ->  ( ps  ->  A  =  B ) )
32con3d 620 . 2  |-  ( ph  ->  ( -.  A  =  B  ->  -.  ps )
)
41, 3syl5bi 151 1  |-  ( ph  ->  ( A  =/=  B  ->  -.  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    =/= wne 2308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-in1 603  ax-in2 604
This theorem depends on definitions:  df-bi 116  df-ne 2309
This theorem is referenced by:  necon3d  2352  disjnim  3920  fodjumkvlemres  7033  nlt1pig  7156  eucalglt  11744  nprm  11810  0nnei  12331
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