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Theorem preq1d 3520
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 3514 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
31, 2syl 14 1 (𝜑 → {𝐴, 𝐶} = {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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