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Theorem ceqsex4v 2614
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 → (𝜑𝜓))
ceqsex4v.8 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex4v.9 (𝑧 = 𝐶 → (𝜒𝜃))
ceqsex4v.10 (𝑤 = 𝐷 → (𝜃𝜏))
Assertion
Ref Expression
ceqsex4v (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 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
StepHypRef Expression
1 19.42vv 1804 . . . 4 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷𝜑)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)))
2 3anass 900 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ ((𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑)))
3 df-3an 898 . . . . . . 7 ((𝑧 = 𝐶𝑤 = 𝐷𝜑) ↔ ((𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑))
43anbi2i 438 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷𝜑)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ ((𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑)))
52, 4bitr4i 180 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷𝜑)))
652exbii 1513 . . . 4 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ ∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷𝜑)))
7 df-3an 898 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵 ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)))
81, 6, 73bitr4i 205 . . 3 (∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ (𝑥 = 𝐴𝑦 = 𝐵 ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)))
982exbii 1513 . 2 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵 ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)))
10 ceqsex4v.1 . . 3 𝐴 ∈ V
11 ceqsex4v.2 . . 3 𝐵 ∈ V
12 ceqsex4v.7 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
13123anbi3d 1224 . . . 4 (𝑥 = 𝐴 → ((𝑧 = 𝐶𝑤 = 𝐷𝜑) ↔ (𝑧 = 𝐶𝑤 = 𝐷𝜓)))
14132exbidv 1764 . . 3 (𝑥 = 𝐴 → (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑) ↔ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜓)))
15 ceqsex4v.8 . . . . 5 (𝑦 = 𝐵 → (𝜓𝜒))
16153anbi3d 1224 . . . 4 (𝑦 = 𝐵 → ((𝑧 = 𝐶𝑤 = 𝐷𝜓) ↔ (𝑧 = 𝐶𝑤 = 𝐷𝜒)))
17162exbidv 1764 . . 3 (𝑦 = 𝐵 → (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜓) ↔ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜒)))
1810, 11, 14, 17ceqsex2v 2612 . 2 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵 ∧ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜑)) ↔ ∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜒))
19 ceqsex4v.3 . . 3 𝐶 ∈ V
20 ceqsex4v.4 . . 3 𝐷 ∈ V
21 ceqsex4v.9 . . 3 (𝑧 = 𝐶 → (𝜒𝜃))
22 ceqsex4v.10 . . 3 (𝑤 = 𝐷 → (𝜃𝜏))
2319, 20, 21, 22ceqsex2v 2612 . 2 (∃𝑧𝑤(𝑧 = 𝐶𝑤 = 𝐷𝜒) ↔ 𝜏)
249, 18, 233bitri 199 1 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ 𝜑) ↔ 𝜏)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  ceqsex8v  2616
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