Theorem ceqsex3v 2641

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 393	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑)))
2		3anass 923	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)))
3	2	anbi1i 445	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶)) ∧ 𝜑))
4		df-3an 921	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑))
5	4	anbi2i 444	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑)))
6	1, 3, 5	3bitr4i 210	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
7	6	2exbii 1537	. . . 4 ⊢ (∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
8		19.42vv 1829	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
9	7, 8	bitri 182	. . 3 ⊢ (∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)))
10	9	exbii 1536	. 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 1249	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓)))
14	13	2exbidv 1789	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑) ↔ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓)))
15	11, 14	ceqsexv 2638	. . 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 2640	. . 3 ⊢ (∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜓) ↔ 𝜃)
21	15, 20	bitri 182	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦∃𝑧(𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝜑)) ↔ 𝜃)
22	10, 21	bitri 182	1 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284  ∃wex 1421   ∈ wcel 1433  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsex6v  2643