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Theorem preq12 4247
Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Assertion
Ref Expression
preq12 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})

Proof of Theorem preq12
StepHypRef Expression
1 preq1 4245 . 2 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2 preq2 4246 . 2 (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
31, 2sylan9eq 2675 1 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  {cpr 4157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-sn 4156  df-pr 4158
This theorem is referenced by:  preq12i  4250  preq12d  4253  ssprsseq  4332  preq12b  4357  prnebg  4364  snex  4879  relop  5242  opthreg  8475  hashle2pr  13213  wwlktovfo  13651  joinval  16945  meetval  16959  ipole  17098  sylow1  17958  frgpuplem  18125  uspgr2wlkeq  26445  wlkres  26470  wlkp1lem8  26480  usgr2pthlem  26562  2wlkdlem10  26734  1wlkdlem4  26900  3wlkdlem6  26925  3wlkdlem10  26929  imarnf1pr  40628  elsprel  41043  sprsymrelf1lem  41059  sprsymrelf  41063
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