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Theorem preq12 3708
Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Assertion
Ref Expression
preq12  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )

Proof of Theorem preq12
StepHypRef Expression
1 preq1 3706 . 2  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
2 preq2 3707 . 2  |-  ( B  =  D  ->  { C ,  B }  =  { C ,  D }
)
31, 2sylan9eq 2335 1  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   {cpr 3641
This theorem is referenced by:  preq12i  3711  preq12d  3714  preq12b  3788  snex  4216  relop  4834  opthreg  7319  ipole  14261  sylow1  14914  frgpuplem  15081
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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