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Theorem preq12 3686
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 3684 . 2 (𝐴 = 𝐶 → {𝐴, 𝐵} = {𝐶, 𝐵})
2 preq2 3685 . 2 (𝐵 = 𝐷 → {𝐶, 𝐵} = {𝐶, 𝐷})
31, 2sylan9eq 2242 1 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  preq12i  3689  preq12d  3692  preq12b  3785  opthreg  4573  relop  4795  qtopbasss  14478
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