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Theorem neeq12d 2380
Description: Deduction for inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
neeq1d.1  |-  ( ph  ->  A  =  B )
neeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
neeq12d  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )

Proof of Theorem neeq12d
StepHypRef Expression
1 neeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21neeq1d 2378 . 2  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  C ) )
3 neeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43neeq2d 2379 . 2  |-  ( ph  ->  ( B  =/=  C  <->  B  =/=  D ) )
52, 4bitrd 188 1  |-  ( ph  ->  ( A  =/=  C  <->  B  =/=  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    =/= wne 2360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-ne 2361
This theorem is referenced by:  3netr3d  2392  3netr4d  2393  exmidapne  7288  ennnfonelemim  12474  ctinfom  12478  isnzr  13528  apdiff  15250
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