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Theorem ceqsex3v 3460
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 → (𝜑𝜓))
ceqsex3v.5 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex3v.6 (𝑧 = 𝐶 → (𝜒𝜃))
Assertion
Ref Expression
ceqsex3v (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 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
StepHypRef Expression
1 anass 472 . . . . . 6 (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
2 3anass 1097 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)))
32anbi1i 627 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑))
4 df-3an 1091 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑))
54anbi2i 626 . . . . . 6 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
61, 3, 53bitr4i 306 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
762exbii 1856 . . . 4 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
8 19.42vv 1966 . . . 4 (∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
97, 8bitri 278 . . 3 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
109exbii 1855 . 2 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
11 ceqsex3v.1 . . . 4 𝐴 ∈ V
12 ceqsex3v.4 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
13123anbi3d 1444 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ (𝑦 = 𝐵𝑧 = 𝐶𝜓)))
14132exbidv 1932 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓)))
1511, 14ceqsexv 3455 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓))
16 ceqsex3v.2 . . . 4 𝐵 ∈ V
17 ceqsex3v.3 . . . 4 𝐶 ∈ V
18 ceqsex3v.5 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
19 ceqsex3v.6 . . . 4 (𝑧 = 𝐶 → (𝜒𝜃))
2016, 17, 18, 19ceqsex2v 3459 . . 3 (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓) ↔ 𝜃)
2115, 20bitri 278 . 2 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ 𝜃)
2210, 21bitri 278 1 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  Vcvv 3408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-ex 1788  df-clel 2816
This theorem is referenced by:  ceqsex6v  3462
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