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Theorem eqvinop 4779
Description: A variable introduction law for ordered pairs. Analogue of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
eqvinop.1 𝐵 ∈ V
eqvinop.2 𝐶 ∈ V
Assertion
Ref Expression
eqvinop (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem eqvinop
StepHypRef Expression
1 eqvinop.1 . . . . . . . 8 𝐵 ∈ V
2 eqvinop.2 . . . . . . . 8 𝐶 ∈ V
31, 2opth2 4773 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐶))
43anbi2i 725 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)))
5 ancom 464 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6 anass 678 . . . . . 6 (((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
74, 5, 63bitri 284 . . . . 5 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
87exbii 1752 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
9 19.42v 1868 . . . 4 (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
10 opeq2 4239 . . . . . . 7 (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
1110eqeq2d 2524 . . . . . 6 (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
122, 11ceqsexv 3119 . . . . 5 (∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
1312anbi2i 725 . . . 4 ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
148, 9, 133bitri 284 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
1514exbii 1752 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
16 opeq1 4238 . . . 4 (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
1716eqeq2d 2524 . . 3 (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
181, 17ceqsexv 3119 . 2 (∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
1915, 18bitr2i 263 1 (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1938  Vcvv 3077  cop 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035
This theorem is referenced by:  copsexg  4780  ralxpf  5082  oprabid  6458
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