Theorem cotsexgw 5463

Description: Substitution of class 𝐴 for ordered triple ⟨𝑥, 𝑦, 𝑧⟩, analogous to copsexgw 5460. (Contributed by BTernaryTau, 8-Sep-2026.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem cotsexgw

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3455	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3455	. . . 4 ⊢ 𝑦 ∈ V
3		vex 3455	. . . 4 ⊢ 𝑧 ∈ V
4	1, 2, 3	eqvinot 5457	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ↔ ∃𝑢∃𝑣∃𝑤(𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ ∧ ⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩))
5		19.8a 2218	. . . . . . . . . . 11 ⊢ ((⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
6	5	19.8ad 2219	. . . . . . . . . 10 ⊢ ((⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
7	6	19.8ad 2219	. . . . . . . . 9 ⊢ ((⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))
8	7	ex 418	. . . . . . . 8 ⊢ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 → ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
9		vex 3455	. . . . . . . . . 10 ⊢ 𝑢 ∈ V
10		vex 3455	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
11		vex 3455	. . . . . . . . . 10 ⊢ 𝑤 ∈ V
12	9, 10, 11	otth 5453	. . . . . . . . 9 ⊢ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ↔ (𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧))
13	12	anbi1i 636	. . . . . . . . . . 11 ⊢ ((⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) ↔ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑))
14	13	3exbii 1883	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑))
15		3an4anass 1122	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ (𝑤 = 𝑧 ∧ 𝜑)))
16	15	exbii 1881	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ ∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ (𝑤 = 𝑧 ∧ 𝜑)))
17		19.42v 1986	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ (𝑤 = 𝑧 ∧ 𝜑)) ↔ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)))
18	16, 17	bitri 278	. . . . . . . . . . . . . 14 ⊢ (∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)))
19	18	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑦∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ ∃𝑦((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)))
20		anass 474	. . . . . . . . . . . . . 14 ⊢ (((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)) ↔ (𝑢 = 𝑥 ∧ (𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
21	20	exbii 1881	. . . . . . . . . . . . 13 ⊢ (∃𝑦((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)) ↔ ∃𝑦(𝑢 = 𝑥 ∧ (𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
22		19.42v 1986	. . . . . . . . . . . . 13 ⊢ (∃𝑦(𝑢 = 𝑥 ∧ (𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))) ↔ (𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
23	19, 21, 22	3bitri 300	. . . . . . . . . . . 12 ⊢ (∃𝑦∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ (𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
24	23	exbii 1881	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) ↔ ∃𝑥(𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
25		ax12ev2c 2217	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (∃𝑥(𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))) → ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))))
26		ax12ev2c 2217	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → (∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)) → ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)))
27		ax12ev2c 2217	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑧 → (∃𝑧(𝑤 = 𝑧 ∧ 𝜑) → 𝜑))
28	26, 27	sylan9 517	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑦 ∧ 𝑤 = 𝑧) → (∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑)) → 𝜑))
29	25, 28	sylan9 517	. . . . . . . . . . . 12 ⊢ ((𝑢 = 𝑥 ∧ (𝑣 = 𝑦 ∧ 𝑤 = 𝑧)) → (∃𝑥(𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))) → 𝜑))
30	29	3impb 1132	. . . . . . . . . . 11 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) → (∃𝑥(𝑢 = 𝑥 ∧ ∃𝑦(𝑣 = 𝑦 ∧ ∃𝑧(𝑤 = 𝑧 ∧ 𝜑))) → 𝜑))
31	24, 30	biimtrid 245	. . . . . . . . . 10 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) → (∃𝑥∃𝑦∃𝑧((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) ∧ 𝜑) → 𝜑))
32	14, 31	biimtrid 245	. . . . . . . . 9 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦 ∧ 𝑤 = 𝑧) → (∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → 𝜑))
33	12, 32	sylbi 220	. . . . . . . 8 ⊢ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → (∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) → 𝜑))
34	8, 33	impbid 215	. . . . . . 7 ⊢ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
35		eqeq1 2765	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ↔ ⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩))
36	35	anbi1d 643	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) ↔ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
37	36	3exbidv 1958	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))
38	37	bibi2d 345	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → ((𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)) ↔ (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
39	35, 38	imbi12d 347	. . . . . . 7 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → ((𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))) ↔ (⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))))
40	34, 39	mpbiri 261	. . . . . 6 ⊢ (𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
41	40	adantr 486	. . . . 5 ⊢ ((𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ ∧ ⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
42	41	exlimiv 1963	. . . 4 ⊢ (∃𝑤(𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ ∧ ⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
43	42	exlimivv 1965	. . 3 ⊢ (∃𝑢∃𝑣∃𝑤(𝐴 = ⟨𝑢, 𝑣, 𝑤⟩ ∧ ⟨𝑢, 𝑣, 𝑤⟩ = ⟨𝑥, 𝑦, 𝑧⟩) → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
44	4, 43	sylbi 220	. 2 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑))))
45	44	pm2.43i 53	1 ⊢ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ 𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812  ⟨cotp 4592

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 2733  ax-sep 5249  ax-pr 5391

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-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593

This theorem is used by:  mosubott  5484