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Theorem opeqex 3977
Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
Assertion
Ref Expression
opeqex (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ((A V B V) ↔ (𝐶 V 𝐷 V)))

Proof of Theorem opeqex
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (x A, B⟩ ↔ x 𝐶, 𝐷⟩))
21exbidv 1703 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (x x A, B⟩ ↔ x x 𝐶, 𝐷⟩))
3 opm 3962 . 2 (x x A, B⟩ ↔ (A V B V))
4 opm 3962 . 2 (x x 𝐶, 𝐷⟩ ↔ (𝐶 V 𝐷 V))
52, 3, 43bitr3g 211 1 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ((A V B V) ↔ (𝐶 V 𝐷 V)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   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
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:  epelg  4018
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