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Theorem preq12 3721
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 3719 . 2  |-  ( A  =  C  ->  { A ,  B }  =  { C ,  B }
)
2 preq2 3720 . 2  |-  ( B  =  D  ->  { C ,  B }  =  { C ,  D }
)
31, 2sylan9eq 2348 1  |-  ( ( A  =  C  /\  B  =  D )  ->  { A ,  B }  =  { C ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   {cpr 3654
This theorem is referenced by:  preq12i  3724  preq12d  3727  preq12b  3804  snex  4232  relop  4850  opthreg  7335  ipole  14277  sylow1  14930  frgpuplem  15097  3v3e3cycl1  28390  4cycl4v4e  28412  4cycl4dv4e  28414
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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