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Theorem opcom 3978
Description: An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
Hypotheses
Ref Expression
opcom.1 A V
opcom.2 B V
Assertion
Ref Expression
opcom (⟨A, B⟩ = ⟨B, A⟩ ↔ A = B)

Proof of Theorem opcom
StepHypRef Expression
1 opcom.1 . . 3 A V
2 opcom.2 . . 3 B V
31, 2opth 3965 . 2 (⟨A, B⟩ = ⟨B, A⟩ ↔ (A = B B = A))
4 eqcom 2039 . . 3 (B = AA = B)
54anbi2i 430 . 2 ((A = B B = A) ↔ (A = B A = B))
6 anidm 376 . 2 ((A = B A = B) ↔ A = B)
73, 5, 63bitri 195 1 (⟨A, B⟩ = ⟨B, A⟩ ↔ A = B)
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  cop 3370
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  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-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by: (None)
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