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Theorem 3netr3d 2317
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.)
Hypotheses
Ref Expression
3netr3d.1  |-  ( ph  ->  A  =/=  B )
3netr3d.2  |-  ( ph  ->  A  =  C )
3netr3d.3  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
3netr3d  |-  ( ph  ->  C  =/=  D )

Proof of Theorem 3netr3d
StepHypRef Expression
1 3netr3d.1 . 2  |-  ( ph  ->  A  =/=  B )
2 3netr3d.2 . . 3  |-  ( ph  ->  A  =  C )
3 3netr3d.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3neeq12d 2305 . 2  |-  ( ph  ->  ( A  =/=  B  <->  C  =/=  D ) )
51, 4mpbid 146 1  |-  ( ph  ->  C  =/=  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    =/= wne 2285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-ne 2286
This theorem is referenced by: (None)
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