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Theorem preq12d 3446
Description: Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)
Hypotheses
Ref Expression
preq1d.1 (φA = B)
preq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
preq12d (φ → {A, 𝐶} = {B, 𝐷})

Proof of Theorem preq12d
StepHypRef Expression
1 preq1d.1 . 2 (φA = B)
2 preq12d.2 . 2 (φ𝐶 = 𝐷)
3 preq12 3440 . 2 ((A = B 𝐶 = 𝐷) → {A, 𝐶} = {B, 𝐷})
41, 2, 3syl2anc 391 1 (φ → {A, 𝐶} = {B, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   = 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-bnd 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:  opeq1  3540  opeq2  3541
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