Theorem copsexgw 5458

Description: Version of copsexg 5460 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by GG, 26-Jan-2024.) Shorten proof and remove dependency on ax-10 2178. (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 3454	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
3	1, 2	eqvinop 5455	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
4		19.8a 2217	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
5	4	19.8ad 2218	. . . . . . . 8 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	ex 418	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 → ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
7		vex 3454	. . . . . . . . 9 ⊢ 𝑧 ∈ V
8		vex 3454	. . . . . . . . 9 ⊢ 𝑤 ∈ V
9	7, 8	opth 5444	. . . . . . . 8 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	anbi1i 636	. . . . . . . . . 10 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
11	10	2exbii 1882	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑))
12		anass 474	. . . . . . . . . . . . 13 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
13	12	exbii 1881	. . . . . . . . . . . 12 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)))
14		19.42v 1986	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ 𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
15	13, 14	bitri 278	. . . . . . . . . . 11 ⊢ (∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
16	15	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
17		euequ 2622	. . . . . . . . . . . . . 14 ⊢ ∃!𝑥 𝑥 = 𝑧
18		equcom 2051	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
19	18	eubii 2610	. . . . . . . . . . . . . 14 ⊢ (∃!𝑥 𝑥 = 𝑧 ↔ ∃!𝑥 𝑧 = 𝑥)
20	17, 19	mpbi 233	. . . . . . . . . . . . 13 ⊢ ∃!𝑥 𝑧 = 𝑥
21		eupick 2658	. . . . . . . . . . . . 13 ⊢ ((∃!𝑥 𝑧 = 𝑥 ∧ ∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑))) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
22	20, 21	mpan 703	. . . . . . . . . . . 12 ⊢ (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑧 = 𝑥 → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
23	22	com12 33	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)))
24		euequ 2622	. . . . . . . . . . . . . 14 ⊢ ∃!𝑦 𝑦 = 𝑤
25		equcom 2051	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 ↔ 𝑤 = 𝑦)
26	25	eubii 2610	. . . . . . . . . . . . . 14 ⊢ (∃!𝑦 𝑦 = 𝑤 ↔ ∃!𝑦 𝑤 = 𝑦)
27	24, 26	mpbi 233	. . . . . . . . . . . . 13 ⊢ ∃!𝑦 𝑤 = 𝑦
28		eupick 2658	. . . . . . . . . . . . 13 ⊢ ((∃!𝑦 𝑤 = 𝑦 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → (𝑤 = 𝑦 → 𝜑))
29	27, 28	mpan 703	. . . . . . . . . . . 12 ⊢ (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → (𝑤 = 𝑦 → 𝜑))
30	29	com12 33	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (∃𝑦(𝑤 = 𝑦 ∧ 𝜑) → 𝜑))
31	23, 30	sylan9 517	. . . . . . . . . 10 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥(𝑧 = 𝑥 ∧ ∃𝑦(𝑤 = 𝑦 ∧ 𝜑)) → 𝜑))
32	16, 31	biimtrid 245	. . . . . . . . 9 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) → 𝜑))
33	11, 32	biimtrid 245	. . . . . . . 8 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
34	9, 33	sylbi 220	. . . . . . 7 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) → 𝜑))
35	6, 34	impbid 215	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
36		eqeq1 2764	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩))
37	36	anbi1d 643	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
38	37	2exbidv 1957	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
39	38	bibi2d 345	. . . . . . 7 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
40	36, 39	imbi12d 347	. . . . . 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 261	. . . . 5 ⊢ (𝐴 = ⟨𝑧, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
42	41	adantr 486	. . . 4 ⊢ ((𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
43	42	exlimivv 1965	. . 3 ⊢ (∃𝑧∃𝑤(𝐴 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩) → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
44	3, 43	sylbi 220	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))))
45	44	pm2.43i 53	1 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  ∃!weu 2593  ⟨cop 4589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590

This theorem is used by:  copsex2t  5461  mosubopt  5479  opabidw  5494  brabgaf  33133  copsex2d  37980