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Theorem preq1d 3834
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
preq1d  |-  ( ph  ->  { A ,  C }  =  { B ,  C } )

Proof of Theorem preq1d
StepHypRef Expression
1 preq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 preq1 3828 . 2  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
31, 2syl 16 1  |-  ( ph  ->  { A ,  C }  =  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   {cpr 3760
This theorem is referenced by:  opthwiener  4401  fprg  5856  dfac2  7946  2pthoncl  21453  eupath2lem3  21551  fprb  25155  wopprc  26794  frgraunss  27750  frgra1v  27753  frgra2v  27754  frgra3v  27757  n4cyclfrgra  27773
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 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-un 3270  df-sn 3765  df-pr 3766
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