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Theorem preq12 3821
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 3819 . 2  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
2 preq2 3820 . 2  |-  ( B  =  D  ->  { C ,  B }  =  { C ,  D }
)
31, 2sylan9eq 2432 1  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   {cpr 3751
This theorem is referenced by:  preq12i  3824  preq12d  3827  preq12b  3909  prnebg  3914  snex  4339  relop  4956  opthreg  7499  ipole  14504  sylow1  15157  frgpuplem  15324  sizeusglecusglem1  21352  3v3e3cycl1  21472  4cycl4v4e  21494  4cycl4dv4e  21496
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-un 3261  df-sn 3756  df-pr 3757
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