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Theorem preq2d 3610
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
preq2d  |-  ( ph  ->  { C ,  A }  =  { C ,  B } )

Proof of Theorem preq2d
StepHypRef Expression
1 preq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 preq2 3604 . 2  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
31, 2syl 14 1  |-  ( ph  ->  { C ,  A }  =  { C ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {cpr 3528
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534
This theorem is referenced by:  opthreg  4474
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