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Theorem preq12i 3890
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 3887 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3mp2an 655 1  |-  { A ,  C }  =  { B ,  D }
Colors of variables: wff set class
Syntax hints:    = wceq 1653   {cpr 3817
This theorem is referenced by:  grpbasex  13574  grpplusgx  13575  indistpsx  17076  lgsdir2lem5  21113  wlkntrllem2  21562  tgrpset  31604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823
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