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Theorem preq12d 3644
 Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1d.1 (𝜑𝐴 = 𝐵)
preq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
preq12d (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})

Proof of Theorem preq12d
StepHypRef Expression
1 preq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 preq12d.2 . 2 (𝜑𝐶 = 𝐷)
3 preq12 3638 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → {𝐴, 𝐶} = {𝐵, 𝐷})
41, 2, 3syl2anc 409 1 (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐷})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1335  {cpr 3561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567 This theorem is referenced by:  opeq1  3741  opeq2  3742  xrminrecl  11163  xrminadd  11165  xmetxp  12878  xmetxpbl  12879  txmetcnp  12889
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