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Theorem neleq12d 25036
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 2550 . . 3  |-  ( A  =  B  ->  ( A  e/  C  <->  B  e/  C ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( A  e/  C  <->  B  e/  C ) )
4 neleq12d.2 . . 3  |-  ( ph  ->  C  =  D )
5 neleq2 2551 . . 3  |-  ( C  =  D  ->  ( B  e/  C  <->  B  e/  D ) )
64, 5syl 15 . 2  |-  ( ph  ->  ( B  e/  C  <->  B  e/  D ) )
73, 6bitrd 244 1  |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e/ wnel 2460
This theorem is referenced by:  isibg2  26213  isibg2aa  26215  isibg2aalem1  26216  isibg2aalem2  26217  nbgranself  28283  nbgrassovt  28284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292  df-nel 2462
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