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Theorem preq12i 3524
Description: Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1i.1  |-  A  =  B
preq12i.2  |-  C  =  D
Assertion
Ref Expression
preq12i  |-  { A ,  C }  =  { B ,  D }

Proof of Theorem preq12i
StepHypRef Expression
1 preq1i.1 . 2  |-  A  =  B
2 preq12i.2 . 2  |-  C  =  D
3 preq12 3521 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3mp2an 417 1  |-  { A ,  C }  =  { B ,  D }
Colors of variables: wff set class
Syntax hints:    = wceq 1289   {cpr 3447
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453
This theorem is referenced by: (None)
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