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Theorem syl5eqner 2753
Description: A chained equality inference for inequality. (Contributed by NM, 6-Jun-2012.) (Proof shortened by Wolf Lammen, 19-Nov-2019.)
Hypotheses
Ref Expression
syl5eqner.1  |-  B  =  A
syl5eqner.2  |-  ( ph  ->  B  =/=  C )
Assertion
Ref Expression
syl5eqner  |-  ( ph  ->  A  =/=  C )

Proof of Theorem syl5eqner
StepHypRef Expression
1 syl5eqner.1 . . 3  |-  B  =  A
21a1i 11 . 2  |-  ( ph  ->  B  =  A )
3 syl5eqner.2 . 2  |-  ( ph  ->  B  =/=  C )
42, 3eqnetrrd 2746 1  |-  ( ph  ->  A  =/=  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    =/= wne 2648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-cleq 2446  df-ne 2650
This theorem is referenced by:  fclsfnflim  19733  ptcmplem2  19758  vieta1lem1  21910  vieta1lem2  21911  signsvfpn  27131  signsvfnn  27132  cdleme3h  34218  cdleme7ga  34231
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