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Theorem preq2 3489
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 3488 . 2 (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2 prcom 3487 . 2 {𝐶, 𝐴} = {𝐴, 𝐶}
3 prcom 3487 . 2 {𝐶, 𝐵} = {𝐵, 𝐶}
41, 2, 33eqtr4g 2140 1 (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  {cpr 3418
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2613  df-un 2987  df-sn 3423  df-pr 3424
This theorem is referenced by:  preq12  3490  preq2i  3492  preq2d  3495  tpeq2  3498  preq12bg  3586  opeq2  3592  uniprg  3637  intprg  3690  prexg  3996  opth  4022  opeqsn  4037  relop  4537  funopg  4987  pr2ne  6606  hashprg  9918  bj-prexg  11006
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