Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsex3v Structured version   Visualization version   GIF version

Theorem ceqsex3v 3219
 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 679 . . . . . 6 (((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
2 3anass 1035 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)))
32anbi1i 727 . . . . . 6 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶)) ∧ 𝜑))
4 df-3an 1033 . . . . . . 7 ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑))
54anbi2i 726 . . . . . 6 ((𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑)))
61, 3, 53bitr4i 291 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
762exbii 1765 . . . 4 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)))
8 19.42vv 1907 . . . 4 (∃𝑦𝑧(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
97, 8bitri 263 . . 3 (∃𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)))
109exbii 1764 . 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 1397 . . . . 5 (𝑥 = 𝐴 → ((𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ (𝑦 = 𝐵𝑧 = 𝐶𝜓)))
14132exbidv 1839 . . . 4 (𝑥 = 𝐴 → (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑) ↔ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓)))
1511, 14ceqsexv 3215 . . 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 3218 . . 3 (∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜓) ↔ 𝜃)
2115, 20bitri 263 . 2 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦𝑧(𝑦 = 𝐵𝑧 = 𝐶𝜑)) ↔ 𝜃)
2210, 21bitri 263 1 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝜑) ↔ 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by:  ceqsex6v  3221
 Copyright terms: Public domain W3C validator