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Theorem copsex2d 34449
Description: Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
Hypotheses
Ref Expression
copsex2d.xph (𝜑 → ∀𝑥𝜑)
copsex2d.yph (𝜑 → ∀𝑦𝜑)
copsex2d.xch (𝜑 → Ⅎ𝑥𝜒)
copsex2d.ych (𝜑 → Ⅎ𝑦𝜒)
copsex2d.exa (𝜑𝐴𝑈)
copsex2d.exb (𝜑𝐵𝑉)
copsex2d.is ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
copsex2d (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem copsex2d
StepHypRef Expression
1 copsex2d.exa . . 3 (𝜑𝐴𝑈)
2 elisset 3482 . . 3 (𝐴𝑈 → ∃𝑥 𝑥 = 𝐴)
31, 2syl 17 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
4 copsex2d.exb . . 3 (𝜑𝐵𝑉)
5 elisset 3482 . . 3 (𝐵𝑉 → ∃𝑦 𝑦 = 𝐵)
64, 5syl 17 . 2 (𝜑 → ∃𝑦 𝑦 = 𝐵)
7 exdistrv 1957 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8 copsex2d.xph . . . 4 (𝜑 → ∀𝑥𝜑)
9 nfe1 2155 . . . . . . 7 𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
109a1i 11 . . . . . 6 (𝜑 → Ⅎ𝑥𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
11 copsex2d.xch . . . . . 6 (𝜑 → Ⅎ𝑥𝜒)
1210, 11nfbid 1904 . . . . 5 (𝜑 → Ⅎ𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
131219.9d 2204 . . . 4 (𝜑 → (∃𝑥(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
14 copsex2d.yph . . . . 5 (𝜑 → ∀𝑦𝜑)
15 nfe1 2155 . . . . . . . . 9 𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
1615a1i 11 . . . . . . . 8 (𝜑 → Ⅎ𝑦𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
178, 16bj-nfexd 34448 . . . . . . 7 (𝜑 → Ⅎ𝑦𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
18 copsex2d.ych . . . . . . 7 (𝜑 → Ⅎ𝑦𝜒)
1917, 18nfbid 1904 . . . . . 6 (𝜑 → Ⅎ𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
201919.9d 2204 . . . . 5 (𝜑 → (∃𝑦(∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
21 opeq12 4778 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
22 copsexgw 5354 . . . . . . . . . . 11 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ ∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2322bicomd 226 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
2423eqcoms 2829 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
2521, 24syl 17 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
2625adantl 485 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
27 copsex2d.is . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
2826, 27bitrd 282 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
2928ex 416 . . . . 5 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
3014, 20, 29bj-exlimd 33966 . . . 4 (𝜑 → (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
318, 13, 30bj-exlimd 33966 . . 3 (𝜑 → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
327, 31syl5bir 246 . 2 (𝜑 → ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
333, 6, 32mp2and 698 1 (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wnf 1785  wcel 2115  cop 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547
This theorem is referenced by:  copsex2b  34450  opelopabd  34451
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