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Theorem neleq12d 24285
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 2510 . . 3  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
31, 2syl 17 . 2  |-  ( ph  ->  ( A  e/  C  <->  B  e/  C ) )
4 neleq12d.2 . . 3  |-  ( ph  ->  C  =  D )
5 neleq2 2511 . . 3  |-  ( C  =  D  ->  ( B  e/  C  <->  B  e/  D ) )
64, 5syl 17 . 2  |-  ( ph  ->  ( B  e/  C  <->  B  e/  D ) )
73, 6bitrd 246 1  |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e/ wnel 2420
This theorem is referenced by:  isibg2  25463  isibg2aa  25465  isibg2aalem1  25466  isibg2aalem2  25467
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628  ax-4 1692  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-cleq 2249  df-clel 2252  df-nel 2422
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