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Theorem preq12d 3714
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1d.1  |-  ( ph  ->  A  =  B )
preq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
preq12d  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )

Proof of Theorem preq12d
StepHypRef Expression
1 preq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 preq12d.2 . 2  |-  ( ph  ->  C  =  D )
3 preq12 3708 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  { A ,  C }  =  { B ,  D } )
41, 2, 3syl2anc 642 1  |-  ( ph  ->  { A ,  C }  =  { B ,  D } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {cpr 3641
This theorem is referenced by:  opeq1  3796  f1oprswap  5515  wunex2  8360  wuncval2  8369  prdsval  13355  imasval  13414  ipoval  14257  frmdval  14473  tmsval  18027  tmsxpsval  18084  uniiccdif  18933  dchrval  20473  disjdifprg  23352  kur14lem1  23737  kur14  23747  iseupa  23881  eupaseg  23888  eupares  23899  eupap1  23900  eupath2lem3  23903  dfac21  27164  mendval  27491  s4prop  28090  frgra2v  28177  frgra3vlem1  28178  frgra3vlem2  28179  tgrpfset  30933  tgrpset  30934  hlhilset  32127
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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