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Theorem preq1d 4218
 Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypothesis
Ref Expression
preq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
preq1d (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})

Proof of Theorem preq1d
StepHypRef Expression
1 preq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 preq1 4212 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
31, 2syl 17 1 (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  propeqop  4895  opthwiener  4901  fprg  6327  fnpr2g  6379  dfac2  8836  symg2bas  17641  2pthoncl  26133  wwlknredwwlkn  26254  wwlkextprop  26272  clwwlkgt0  26299  clwlkisclwwlklem2fv1  26310  clwlkisclwwlklem2fv2  26311  clwlkisclwwlklem2a  26313  clwlkisclwwlklem0  26316  clwwisshclwwlem  26334  eupath2lem3  26506  frgraunss  26522  frgra1v  26525  frgra2v  26526  frgra3v  26529  n4cyclfrgra  26545  fprb  30916  wopprc  36615  crctcsh1wlkn0lem6  41018  wwlksnredwwlkn  41101  wwlksnextprop  41118  clwlkclwwlklem2fv1  41204  clwlkclwwlklem2fv2  41205  clwlkclwwlklem2a  41207  clwlkclwwlklem3  41210  clwwlks1loop  41215  clwwlksn1loop  41216  clwwisshclwwslem  41234  frcond1  41438  frgr1v  41441  nfrgr2v  41442  frgr3v  41445  n4cyclfrgr  41461
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