Theorem preq2 3448

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

Step	Hyp	Ref	Expression
1		preq1 3447	. 2 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐶} = {𝐵, 𝐶})
2		prcom 3446	. 2 ⊢ {𝐶, 𝐴} = {𝐴, 𝐶}
3		prcom 3446	. 2 ⊢ {𝐶, 𝐵} = {𝐵, 𝐶}
4	1, 2, 3	3eqtr4g 2097	1 ⊢ (𝐴 = 𝐵 → {𝐶, 𝐴} = {𝐶, 𝐵})

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:  preq12  3449  preq2i  3451  preq2d  3454  tpeq2  3457  preq12bg  3544  opeq2  3550  uniprg  3595  intprg  3648  prexgOLD  3946  prexg  3947  opth  3974  opeqsn  3989  relop  4486  funopg  4934  bj-prexg  10031