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Theorem eqvinop 3980
 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 3977 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐶))
43anbi2i 430 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)))
5 ancom 253 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6 anass 381 . . . . . 6 (((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
74, 5, 63bitri 195 . . . . 5 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
87exbii 1496 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
9 19.42v 1786 . . . 4 (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
10 opeq2 3550 . . . . . . 7 (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
1110eqeq2d 2051 . . . . . 6 (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
122, 11ceqsexv 2593 . . . . 5 (∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
1312anbi2i 430 . . . 4 ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
148, 9, 133bitri 195 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
1514exbii 1496 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
16 opeq1 3549 . . . 4 (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
1716eqeq2d 2051 . . 3 (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
181, 17ceqsexv 2593 . 2 (∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
1915, 18bitr2i 174 1 (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557  ⟨cop 3378 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384 This theorem is referenced by:  copsexg  3981  ralxpf  4482  rexxpf  4483  oprabid  5537
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