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Theorem preq12 3877
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 3875 . 2  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
2 preq2 3876 . 2  |-  ( B  =  D  ->  { C ,  B }  =  { C ,  D }
)
31, 2sylan9eq 2487 1  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652   {cpr 3807
This theorem is referenced by:  preq12i  3880  preq12d  3883  preq12b  3966  prnebg  3971  snex  4397  relop  5015  opthreg  7565  ipole  14576  sylow1  15229  frgpuplem  15396  sizeusglecusglem1  21485  3v3e3cycl1  21623  4cycl4v4e  21645  4cycl4dv4e  21647  imarnf1pr  28060  usgra2pthlem1  28253  usg2wlkonot  28293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813
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