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Theorem eqvinop 4007
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 4004 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩ ↔ (𝑥 = 𝐵𝑦 = 𝐶))
43anbi2i 438 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)))
5 ancom 257 . . . . . 6 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = 𝐵𝑦 = 𝐶)) ↔ ((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
6 anass 387 . . . . . 6 (((𝑥 = 𝐵𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
74, 5, 63bitri 199 . . . . 5 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
87exbii 1512 . . . 4 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
9 19.42v 1802 . . . 4 (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)))
10 opeq2 3577 . . . . . . 7 (𝑦 = 𝐶 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐶⟩)
1110eqeq2d 2067 . . . . . 6 (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, 𝐶⟩))
122, 11ceqsexv 2610 . . . . 5 (∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩) ↔ 𝐴 = ⟨𝑥, 𝐶⟩)
1312anbi2i 438 . . . 4 ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
148, 9, 133bitri 199 . . 3 (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ (𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
1514exbii 1512 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩) ↔ ∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩))
16 opeq1 3576 . . . 4 (𝑥 = 𝐵 → ⟨𝑥, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
1716eqeq2d 2067 . . 3 (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶⟩ ↔ 𝐴 = ⟨𝐵, 𝐶⟩))
181, 17ceqsexv 2610 . 2 (∃𝑥(𝑥 = 𝐵𝐴 = ⟨𝑥, 𝐶⟩) ↔ 𝐴 = ⟨𝐵, 𝐶⟩)
1915, 18bitr2i 178 1 (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝐶⟩))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cop 3405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411
This theorem is referenced by:  copsexg  4008  ralxpf  4509  rexxpf  4510  oprabid  5564
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