Theorem copsex2d 35310

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	⊢ (𝜑 → 𝐴 ∈ 𝑈)
copsex2d.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
copsex2d.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
copsex2d	⊢ (𝜑 → (∃𝑥∃𝑦(⟨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

Step	Hyp	Ref	Expression
1		copsex2d.exa	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		elisset 2820	. . 3 ⊢ (𝐴 ∈ 𝑈 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
4		copsex2d.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
5		elisset 2820	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐵)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 = 𝐵)
7		exdistrv 1959	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		copsex2d.xph	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
9		nfe1 2147	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
10	9	a1i 11	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
11		copsex2d.xch	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝜒)
12	10, 11	nfbid 1905	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
13	12	19.9d 2196	. . . 4 ⊢ (𝜑 → (∃𝑥(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
14		copsex2d.yph	. . . . 5 ⊢ (𝜑 → ∀𝑦𝜑)
15		nfe1 2147	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
16	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑦∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
17	8, 16	bj-nfexd 35309	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
18		copsex2d.ych	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
19	17, 18	nfbid 1905	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
20	19	19.9d 2196	. . . . 5 ⊢ (𝜑 → (∃𝑦(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
21		opeq12 4806	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
22		copsexgw 5404	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
23	22	bicomd 222	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
24	23	eqcoms 2746	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
25	21, 24	syl 17	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
26	25	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
27		copsex2d.is	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
28	26, 27	bitrd 278	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
29	28	ex 413	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
30	14, 20, 29	bj-exlimd 34806	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
31	8, 13, 30	bj-exlimd 34806	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
32	7, 31	syl5bir 242	. 2 ⊢ (𝜑 → ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
33	3, 6, 32	mp2and 696	1 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2106  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by:  copsex2b  35311  opelopabd  35312