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Theorem preq2 3569
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 3568 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2 prcom 3567 . 2 {𝐶, 𝐴} = {𝐴, 𝐶}
3 prcom 3567 . 2 {𝐶, 𝐵} = {𝐵, 𝐶}
41, 2, 33eqtr4g 2173 1 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  {cpr 3496
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by:  preq12  3570  preq2i  3572  preq2d  3575  tpeq2  3578  preq12bg  3668  opeq2  3674  uniprg  3719  intprg  3772  prexg  4101  opth  4127  opeqsn  4142  relop  4657  funopg  5125  pr2ne  7014  hashprg  10494  bj-prexg  12911
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