Theorem ceqsex3v 3545

Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Hypotheses

Ref	Expression
ceqsex3v.1	⊢ 𝐴 ∈ V
ceqsex3v.2	⊢ 𝐵 ∈ V
ceqsex3v.3	⊢ 𝐶 ∈ V
ceqsex3v.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsex3v.5	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ceqsex3v.6	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
ceqsex3v	⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑦,𝑧)   𝜒(𝑥,𝑧)   𝜃(𝑥,𝑦)

Proof of Theorem ceqsex3v

Step	Hyp	Ref	Expression
1		anass 471	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑)))
2		3anass 1091	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)))
3	2	anbi1i 625	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)) ∧ 𝜑))
4		df-3an 1085	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑))
5	4	anbi2i 624	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑)))
6	1, 3, 5	3bitr4i 305	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
7	6	2exbii 1845	. . . 4 ⊢ (∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
8		19.42vv 1954	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
9	7, 8	bitri 277	. . 3 ⊢ (∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
10	9	exbii 1844	. 2 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
11		ceqsex3v.1	. . . 4 ⊢ 𝐴 ∈ V
12		ceqsex3v.4	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	12	3anbi3d 1438	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓)))
14	13	2exbidv 1921	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓)))
15	11, 14	ceqsexv 3541	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓))
16		ceqsex3v.2	. . . 4 ⊢ 𝐵 ∈ V
17		ceqsex3v.3	. . . 4 ⊢ 𝐶 ∈ V
18		ceqsex3v.5	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
19		ceqsex3v.6	. . . 4 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
20	16, 17, 18, 19	ceqsex2v 3544	. . 3 ⊢ (∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓) ↔ 𝜃)
21	15, 20	bitri 277	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ 𝜃)
22	10, 21	bitri 277	1 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1085  df-ex 1777  df-nf 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  ceqsex6v  3547