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Theorem preq2d 3890
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
preq2d  |-  ( ph  ->  { C ,  A }  =  { C ,  B } )

Proof of Theorem preq2d
StepHypRef Expression
1 preq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 preq2 3884 . 2  |-  ( A  =  B  ->  { C ,  A }  =  { C ,  B }
)
31, 2syl 16 1  |-  ( ph  ->  { C ,  A }  =  { C ,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {cpr 3815
This theorem is referenced by:  opeq2  3985  opthwiener  4458  fprg  5915  opthreg  7573  s2prop  11861  indislem  17064  iscon  17476  hmphindis  17829  wilthlem2  20852  ispth  21568  1pthonlem2  21590  2pthoncl  21603  eupath2lem3  21701  eupath2  21702  measxun2  24564  fprb  25397  altopthsn  25806  frgraunss  28385  frgra2v  28389  frgra3v  28392  n4cyclfrgra  28408  mapdindp4  32521
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-sn 3820  df-pr 3821
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