Theorem eqvinot 5457

Description: A variable introduction law for ordered triples, analogous to eqvinop 5456. (Contributed by BTernaryTau, 8-Sep-2026.)

Hypotheses

Ref	Expression
eqvinot.1	⊢ 𝐵 ∈ V
eqvinot.2	⊢ 𝐶 ∈ V
eqvinot.3	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
eqvinot	⊢ (𝐴 = ⟨𝐵, 𝐶, 𝐷⟩ ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧

Proof of Theorem eqvinot

Step	Hyp	Ref	Expression
1		19.42v 1986	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)) ↔ (𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)))
2		19.42v 1986	. . . . . . 7 ⊢ (∃𝑧((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ (𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ ∃𝑧(𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)))
3		vex 3455	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
4		vex 3455	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
5		vex 3455	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
6	3, 4, 5	otth 5453	. . . . . . . . . 10 ⊢ (⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩ ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ∧ 𝑧 = 𝐷))
7	6	anbi2i 635	. . . . . . . . 9 ⊢ ((𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩) ↔ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ∧ 𝑧 = 𝐷)))
8		ancom 466	. . . . . . . . 9 ⊢ ((𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ (𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ∧ 𝑧 = 𝐷)) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ∧ 𝑧 = 𝐷) ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩))
9		3an4anass 1122	. . . . . . . . 9 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶 ∧ 𝑧 = 𝐷) ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ (𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)))
10	7, 8, 9	3bitrri 301	. . . . . . . 8 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ (𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))
11	10	exbii 1881	. . . . . . 7 ⊢ (∃𝑧((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ (𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)) ↔ ∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))
12		eqvinot.3	. . . . . . . . 9 ⊢ 𝐷 ∈ V
13		oteq3 4844	. . . . . . . . . 10 ⊢ (𝑧 = 𝐷 → ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝑥, 𝑦, 𝐷⟩)
14	13	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑧 = 𝐷 → (𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ↔ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩))
15	12, 14	ceqsexv 3499	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩) ↔ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)
16	15	anbi2i 635	. . . . . . 7 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ ∃𝑧(𝑧 = 𝐷 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝑧⟩)) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩))
17	2, 11, 16	3bitr3i 304	. . . . . 6 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩) ↔ ((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩))
18		anass 474	. . . . . 6 ⊢ (((𝑥 = 𝐵 ∧ 𝑦 = 𝐶) ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩) ↔ (𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)))
19	17, 18	bitr2i 279	. . . . 5 ⊢ ((𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)) ↔ ∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))
20	19	exbii 1881	. . . 4 ⊢ (∃𝑦(𝑥 = 𝐵 ∧ (𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))
21		eqvinot.2	. . . . . 6 ⊢ 𝐶 ∈ V
22		oteq2 4843	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ⟨𝑥, 𝑦, 𝐷⟩ = ⟨𝑥, 𝐶, 𝐷⟩)
23	22	eqeq2d 2772	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐴 = ⟨𝑥, 𝑦, 𝐷⟩ ↔ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩))
24	21, 23	ceqsexv 3499	. . . . 5 ⊢ (∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩) ↔ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩)
25	24	anbi2i 635	. . . 4 ⊢ ((𝑥 = 𝐵 ∧ ∃𝑦(𝑦 = 𝐶 ∧ 𝐴 = ⟨𝑥, 𝑦, 𝐷⟩)) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩))
26	1, 20, 25	3bitr3i 304	. . 3 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩) ↔ (𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩))
27	26	exbii 1881	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩) ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩))
28		eqvinot.1	. . 3 ⊢ 𝐵 ∈ V
29		oteq1 4842	. . . 4 ⊢ (𝑥 = 𝐵 → ⟨𝑥, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
30	29	eqeq2d 2772	. . 3 ⊢ (𝑥 = 𝐵 → (𝐴 = ⟨𝑥, 𝐶, 𝐷⟩ ↔ 𝐴 = ⟨𝐵, 𝐶, 𝐷⟩))
31	28, 30	ceqsexv 3499	. 2 ⊢ (∃𝑥(𝑥 = 𝐵 ∧ 𝐴 = ⟨𝑥, 𝐶, 𝐷⟩) ↔ 𝐴 = ⟨𝐵, 𝐶, 𝐷⟩)
32	27, 31	bitr2i 279	1 ⊢ (𝐴 = ⟨𝐵, 𝐶, 𝐷⟩ ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑦, 𝑧⟩ ∧ ⟨𝑥, 𝑦, 𝑧⟩ = ⟨𝐵, 𝐶, 𝐷⟩))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451  ⟨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-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:  cotsexgw  5463