Theorem copsex2d 37666

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 2851	. . 3 ⊢ (𝐴 ∈ 𝑈 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
4		copsex2d.exb	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
5		elisset 2851	. . 3 ⊢ (𝐵 ∈ 𝑉 → ∃𝑦 𝑦 = 𝐵)
6	4, 5	syl 18	. 2 ⊢ (𝜑 → ∃𝑦 𝑦 = 𝐵)
7		exdistrv 1982	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
8		copsex2d.xph	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
9		nfe1 2191	. . . . . . 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 1929	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
13	12	19.9d 2245	. . . 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 2191	. . . . . . . . 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 37663	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
18		copsex2d.ych	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑦𝜒)
19	17, 18	nfbid 1929	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
20	19	19.9d 2245	. . . . 5 ⊢ (𝜑 → (∃𝑦(∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
21		opeq12 4841	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
22		copsexgw 5470	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ ∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
23	22	bicomd 226	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
24	23	eqcoms 2777	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
25	21, 24	syl 18	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜓))
26	25	adantl 486	. . . . . . 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 282	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
29	28	ex 417	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
30	14, 20, 29	bj-exlimd 37115	. . . 4 ⊢ (𝜑 → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
31	8, 13, 30	bj-exlimd 37115	. . 3 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
32	7, 31	biimtrrid 246	. 2 ⊢ (𝜑 → ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒)))
33	3, 6, 32	mp2and 711	1 ⊢ (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149  ⟨cop 4597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598

This theorem is referenced by:  copsex2b  37667  opelopabd  37668