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Theorem eqvinop 4033
Description: A variable introduction law for ordered pairs. Analog 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 4030 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐶))
43anbi2i 445 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)))
5 ancom 262 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6 anass 393 . . . . . 6 (((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
74, 5, 63bitri 204 . . . . 5 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
87exbii 1537 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
9 19.42v 1829 . . . 4 (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
10 opeq2 3597 . . . . . . 7 (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
1110eqeq2d 2094 . . . . . 6 (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
122, 11ceqsexv 2649 . . . . 5 (∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
1312anbi2i 445 . . . 4 ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
148, 9, 133bitri 204 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
1514exbii 1537 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
16 opeq1 3596 . . . 4 (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
1716eqeq2d 2094 . . 3 (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
181, 17ceqsexv 2649 . 2 (∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
1915, 18bitr2i 183 1 (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2612  cop 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431
This theorem is referenced by:  copsexg  4034  ralxpf  4539  rexxpf  4540  oprabid  5615
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