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Theorem neleq12d 2352
Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
Hypotheses
Ref Expression
neleq12d.1  |-  ( ph  ->  A  =  B )
neleq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
neleq12d  |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )

Proof of Theorem neleq12d
StepHypRef Expression
1 neleq12d.1 . . 3  |-  ( ph  ->  A  =  B )
2 neleq1 2350 . . 3  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
31, 2syl 14 . 2  |-  ( ph  ->  ( A  e/  C  <->  B  e/  C ) )
4 neleq12d.2 . . 3  |-  ( ph  ->  C  =  D )
5 neleq2 2351 . . 3  |-  ( C  =  D  ->  ( B  e/  C  <->  B  e/  D ) )
64, 5syl 14 . 2  |-  ( ph  ->  ( B  e/  C  <->  B  e/  D ) )
73, 6bitrd 186 1  |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1287    e/ wnel 2346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-clel 2081  df-nel 2347
This theorem is referenced by: (None)
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