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Theorem preq1d 4667
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 4661 . 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 1528  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-sn 4558  df-pr 4560
This theorem is referenced by:  propeqop  5388  opthwiener  5395  fprg  6909  fprb  6948  fnpr2g  6964  dfac2b  9544  symg2bas  18455  crctcshwlkn0lem6  27520  wwlksnredwwlkn  27600  wwlksnextprop  27618  clwwlk1loop  27693  clwlkclwwlklem2fv1  27700  clwlkclwwlklem2fv2  27701  clwlkclwwlklem2a  27703  clwlkclwwlklem3  27706  clwwisshclwwslem  27719  clwwlknlbonbgr1  27744  clwwlkn1  27746  frcond1  27972  frgr1v  27977  nfrgr2v  27978  frgr3v  27981  n4cyclfrgr  27997  2clwwlk2clwwlklem  28052  wopprc  39505  mnurndlem1  40494  isomuspgrlem2d  43873  rrx2xpref1o  44633
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