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Theorem preq12i 3711
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 3708 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3mp2an 653 1  |-  { A ,  C }  =  { B ,  D }
Colors of variables: wff set class
Syntax hints:    = wceq 1623   {cpr 3641
This theorem is referenced by:  grpbasex  13251  grpplusgx  13252  indistpsx  16747  lgsdir2lem5  20566  tgrpset  30934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647
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