Theorem copsexgw 5432

Description: Version of copsexg 5434 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by GG, 26-Jan-2024.) Shorten proof and remove dependency on ax-10 2147. (Revised by Eric Schmidt, 2-May-2026.)

Assertion

Ref	Expression
copsexgw	⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem copsexgw

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3431	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3431	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqvinop 5429	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
4		19.8a 2188	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	19.8ad 2189	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	ex 412	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
7		vex 3431	. . . . . . . . 9 ⊢ 𝑧 ∈ V
8		vex 3431	. . . . . . . . 9 ⊢ 𝑤 ∈ V
9	7, 8	opth 5418	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	anbi1i 625	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
11	10	2exbii 1851	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
12		anass 468	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
13	12	exbii 1850	. . . . . . . . . . . 12 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
14		19.42v 1955	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
15	13, 14	bitri 275	. . . . . . . . . . 11 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
16	15	exbii 1850	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
17		euequ 2596	. . . . . . . . . . . . . 14 ⊢ ∃!𝑥 𝑥 = 𝑧
18		equcom 2020	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
19	18	eubii 2584	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥 𝑥 = 𝑧 ↔ ∃!𝑥 𝑧 = 𝑥)
20	17, 19	mpbi 230	. . . . . . . . . . . . 13 ⊢ ∃!𝑥 𝑧 = 𝑥
21		eupick 2632	. . . . . . . . . . . . 13 ⊢ ((∃!𝑥 𝑧 = 𝑥 ∧ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
22	20, 21	mpan 691	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
23	22	com12 32	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
24		euequ 2596	. . . . . . . . . . . . . 14 ⊢ ∃!𝑦 𝑦 = 𝑤
25		equcom 2020	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
26	25	eubii 2584	. . . . . . . . . . . . . 14 ⊢ (∃!𝑦 𝑦 = 𝑤 ↔ ∃!𝑦 𝑤 = 𝑦)
27	24, 26	mpbi 230	. . . . . . . . . . . . 13 ⊢ ∃!𝑦 𝑤 = 𝑦
28		eupick 2632	. . . . . . . . . . . . 13 ⊢ ((∃!𝑦 𝑤 = 𝑦 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑤 = 𝑦 → 𝜑))
29	27, 28	mpan 691	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → (𝑤 = 𝑦 → 𝜑))
30	29	com12 32	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → 𝜑))
31	23, 30	sylan9 507	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → 𝜑))
32	16, 31	biimtrid 242	. . . . . . . . 9 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → 𝜑))
33	11, 32	biimtrid 242	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
34	9, 33	sylbi 217	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
35	6, 34	impbid 212	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
36		eqeq1 2739	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
37	36	anbi1d 632	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
38	37	2exbidv 1926	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
39	38	bibi2d 342	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
40	36, 39	imbi12d 344	. . . . . 6 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))))
41	35, 40	mpbiri 258	. . . . 5 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
42	41	adantr 480	. . . 4 ⊢ ((𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
43	42	exlimivv 1934	. . 3 ⊢ (∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
44	3, 43	sylbi 217	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
45	44	pm2.43i 52	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  ∃!weu 2567  ⟨cop 4563

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564

This theorem is referenced by:  copsex2t  5435  mosubopt  5453  opabidw  5468  brabgaf  32667  copsex2d  37441