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Theorem preq1d 3453
 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 3447 . 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 1243  {cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382 This theorem is referenced by: (None)
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