Theorem ceqsex4v 2807

Description: Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Hypotheses

Ref	Expression
ceqsex4v.1	⊢ 𝐴 ∈ V
ceqsex4v.2	⊢ 𝐵 ∈ V
ceqsex4v.3	⊢ 𝐶 ∈ V
ceqsex4v.4	⊢ 𝐷 ∈ V
ceqsex4v.7	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsex4v.8	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ceqsex4v.9	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
ceqsex4v.10	⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
ceqsex4v	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)

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

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

Proof of Theorem ceqsex4v

Step	Hyp	Ref	Expression
1		19.42vv 1926	. . . 4 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
2		3anass 984	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑)))
3		df-3an 982	. . . . . . 7 ⊢ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑) ↔ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑))
4	3	anbi2i 457	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑)))
5	2, 4	bitr4i 187	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
6	5	2exbii 1620	. . . 4 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
7		df-3an 982	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
8	1, 6, 7	3bitr4i 212	. . 3 ⊢ (∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
9	8	2exbii 1620	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)))
10		ceqsex4v.1	. . 3 ⊢ 𝐴 ∈ V
11		ceqsex4v.2	. . 3 ⊢ 𝐵 ∈ V
12		ceqsex4v.7	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
13	12	3anbi3d 1329	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑) ↔ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜓)))
14	13	2exbidv 1882	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑) ↔ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜓)))
15		ceqsex4v.8	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
16	15	3anbi3d 1329	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜓) ↔ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜒)))
17	16	2exbidv 1882	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜓) ↔ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜒)))
18	10, 11, 14, 17	ceqsex2v 2805	. 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜑)) ↔ ∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜒))
19		ceqsex4v.3	. . 3 ⊢ 𝐶 ∈ V
20		ceqsex4v.4	. . 3 ⊢ 𝐷 ∈ V
21		ceqsex4v.9	. . 3 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
22		ceqsex4v.10	. . 3 ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))
23	19, 20, 21, 22	ceqsex2v 2805	. 2 ⊢ (∃𝑧∃𝑤(𝑧 = 𝐶 ∧ 𝑤 = 𝐷 ∧ 𝜒) ↔ 𝜏)
24	9, 18, 23	3bitri 206	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-v 2765

This theorem is referenced by:  ceqsex8v  2809