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Theorem preq1d 3725
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 3719 . 2  |-  ( A  =  B  ->  { A ,  C }  =  { B ,  C }
)
31, 2syl 15 1  |-  ( ph  ->  { A ,  C }  =  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {cpr 3654
This theorem is referenced by:  opthwiener  4284  dfac2  7773  eupath2lem3  23918  fprb  24200  fprg  25236  repcpwti  25264  cbcpcp  25265  pgapspf2  26156  wopprc  27226  frgra1v  28422  frgra2v  28423  frgra3v  28426  n4cyclfrgra  28440
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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