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Theorem preq2 3571
Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
preq2 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})

Proof of Theorem preq2
StepHypRef Expression
1 preq1 3570 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2 prcom 3569 . 2 {𝐶, 𝐴} = {𝐴, 𝐶}
3 prcom 3569 . 2 {𝐶, 𝐵} = {𝐵, 𝐶}
41, 2, 33eqtr4g 2175 1 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  {cpr 3498
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504
This theorem is referenced by:  preq12  3572  preq2i  3574  preq2d  3577  tpeq2  3580  preq12bg  3670  opeq2  3676  uniprg  3721  intprg  3774  prexg  4103  opth  4129  opeqsn  4144  relop  4659  funopg  5127  pr2ne  7016  hashprg  10522  bj-prexg  13036
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