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Theorem preq12 3440
Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Assertion
Ref Expression
preq12 ((A = 𝐶 B = 𝐷) → {A, B} = {𝐶, 𝐷})

Proof of Theorem preq12
StepHypRef Expression
1 preq1 3438 . 2 (A = 𝐶 → {A, B} = {𝐶, B})
2 preq2 3439 . 2 (B = 𝐷 → {𝐶, B} = {𝐶, 𝐷})
31, 2sylan9eq 2089 1 ((A = 𝐶 B = 𝐷) → {A, B} = {𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  preq12i  3443  preq12d  3446  preq12b  3532  opthreg  4234  relop  4429
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