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Theorem preq12i 3724
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 3721 . 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 1632   {cpr 3654
This theorem is referenced by:  grpbasex  13267  grpplusgx  13268  indistpsx  16763  lgsdir2lem5  20582  wlkntrllem4  28348  tgrpset  31556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660
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