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Theorem copsex2b 35238
Description: Biconditional form of copsex2d 35237. TODO: prove a relative version, that is, with 𝑥𝑉𝑦𝑊...(𝐴𝑉𝐵𝑊). (Contributed by BJ, 27-Dec-2023.)
Hypotheses
Ref Expression
copsex2b.xph (𝜑 → ∀𝑥𝜑)
copsex2b.yph (𝜑 → ∀𝑦𝜑)
copsex2b.xch (𝜑 → Ⅎ𝑥𝜒)
copsex2b.ych (𝜑 → Ⅎ𝑦𝜒)
copsex2b.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
copsex2b (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem copsex2b
StepHypRef Expression
1 eqcom 2745 . . . . . . 7 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2 vex 3426 . . . . . . . 8 𝑥 ∈ V
3 vex 3426 . . . . . . . 8 𝑦 ∈ V
42, 3opth 5385 . . . . . . 7 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4bitri 274 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
6 eqvisset 3439 . . . . . . 7 (𝑥 = 𝐴𝐴 ∈ V)
7 eqvisset 3439 . . . . . . 7 (𝑦 = 𝐵𝐵 ∈ V)
86, 7anim12i 612 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
95, 8sylbi 216 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
109adantr 480 . . . 4 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1110exlimivv 1936 . . 3 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1211anim2i 616 . 2 ((𝜑 ∧ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) → (𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
13 simpl 482 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1413anim2i 616 . 2 ((𝜑 ∧ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)) → (𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
15 copsex2b.xph . . . . 5 (𝜑 → ∀𝑥𝜑)
16 ax-5 1914 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V))
1715, 16hban 2300 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ∀𝑥(𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
18 copsex2b.yph . . . . 5 (𝜑 → ∀𝑦𝜑)
19 ax-5 1914 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑦(𝐴 ∈ V ∧ 𝐵 ∈ V))
2018, 19hban 2300 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → ∀𝑦(𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)))
21 copsex2b.xch . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
2221adantr 480 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → Ⅎ𝑥𝜒)
23 copsex2b.ych . . . . 5 (𝜑 → Ⅎ𝑦𝜒)
2423adantr 480 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → Ⅎ𝑦𝜒)
25 simprl 767 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐴 ∈ V)
26 simprr 769 . . . 4 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → 𝐵 ∈ V)
27 copsex2b.is . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
2827adantlr 711 . . . 4 (((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
2917, 20, 22, 24, 25, 26, 28copsex2d 35237 . . 3 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
30 ibar 528 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜒 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
3130adantl 481 . . 3 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (𝜒 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
3229, 31bitrd 278 . 2 ((𝜑 ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
3312, 14, 32pm5.21nd 798 1 (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1537   = wceq 1539  wex 1783  wnf 1787  wcel 2108  Vcvv 3422  cop 4564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565
This theorem is referenced by:  opelopabb  35240
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