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Theorem eqvinop 5381
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 5375 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐶))
43anbi2i 624 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)))
5 ancom 463 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6 anass 471 . . . . . 6 (((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
74, 5, 63bitri 299 . . . . 5 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
87exbii 1847 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
9 19.42v 1953 . . . 4 (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
10 opeq2 4807 . . . . . . 7 (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
1110eqeq2d 2835 . . . . . 6 (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
122, 11ceqsexv 3544 . . . . 5 (∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
1312anbi2i 624 . . . 4 ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
148, 9, 133bitri 299 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
1514exbii 1847 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
16 opeq1 4806 . . . 4 (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
1716eqeq2d 2835 . . 3 (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
181, 17ceqsexv 3544 . 2 (∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
1915, 18bitr2i 278 1 (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1536  wex 1779  wcel 2113  Vcvv 3497  cop 4576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577
This theorem is referenced by:  copsexgw  5384  copsexg  5385  ralxpf  5720  oprabidw  7190  oprabid  7191
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