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Theorem neeq2 2265
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.)
Assertion
Ref Expression
neeq2  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )

Proof of Theorem neeq2
StepHypRef Expression
1 eqeq2 2094 . . 3  |-  ( A  =  B  ->  ( C  =  A  <->  C  =  B ) )
21notbid 625 . 2  |-  ( A  =  B  ->  ( -.  C  =  A  <->  -.  C  =  B ) )
3 df-ne 2252 . 2  |-  ( C  =/=  A  <->  -.  C  =  A )
4 df-ne 2252 . 2  |-  ( C  =/=  B  <->  -.  C  =  B )
52, 3, 43bitr4g 221 1  |-  ( A  =  B  ->  ( C  =/=  A  <->  C  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    = wceq 1287    =/= wne 2251
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-4 1443  ax-17 1462  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-cleq 2078  df-ne 2252
This theorem is referenced by:  neeq2i  2267  neeq2d  2270  fodjuomnilemdc  6743  xrlttri3  9199
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