Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsex8v GIF version

Theorem ceqsex8v 2599
 Description: Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
Hypotheses
Ref Expression
ceqsex8v.1 𝐴 ∈ V
ceqsex8v.2 𝐵 ∈ V
ceqsex8v.3 𝐶 ∈ V
ceqsex8v.4 𝐷 ∈ V
ceqsex8v.5 𝐸 ∈ V
ceqsex8v.6 𝐹 ∈ V
ceqsex8v.7 𝐺 ∈ V
ceqsex8v.8 𝐻 ∈ V
ceqsex8v.9 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsex8v.10 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex8v.11 (𝑧 = 𝐶 → (𝜒𝜃))
ceqsex8v.12 (𝑤 = 𝐷 → (𝜃𝜏))
ceqsex8v.13 (𝑣 = 𝐸 → (𝜏𝜂))
ceqsex8v.14 (𝑢 = 𝐹 → (𝜂𝜁))
ceqsex8v.15 (𝑡 = 𝐺 → (𝜁𝜎))
ceqsex8v.16 (𝑠 = 𝐻 → (𝜎𝜌))
Assertion
Ref Expression
ceqsex8v (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐸,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐹,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐻,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧   𝜏,𝑤   𝜂,𝑣   𝜁,𝑢   𝜎,𝑡   𝜌,𝑠
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠)   𝜓(𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠)   𝜒(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠)   𝜃(𝑥,𝑦,𝑤,𝑣,𝑢,𝑡,𝑠)   𝜏(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝜂(𝑥,𝑦,𝑧,𝑤,𝑢,𝑡,𝑠)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑠)   𝜎(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑠)   𝜌(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡)

Proof of Theorem ceqsex8v
StepHypRef Expression
1 19.42vvvv 1790 . . . . 5 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
2 3anass 889 . . . . . . . 8 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑)))
3 df-3an 887 . . . . . . . . 9 (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑))
43anbi2i 430 . . . . . . . 8 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑)))
52, 4bitr4i 176 . . . . . . 7 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
652exbii 1497 . . . . . 6 (∃𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
762exbii 1497 . . . . 5 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
8 df-3an 887 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
91, 7, 83bitr4i 201 . . . 4 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
1092exbii 1497 . . 3 (∃𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
11102exbii 1497 . 2 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
12 ceqsex8v.1 . . . 4 𝐴 ∈ V
13 ceqsex8v.2 . . . 4 𝐵 ∈ V
14 ceqsex8v.3 . . . 4 𝐶 ∈ V
15 ceqsex8v.4 . . . 4 𝐷 ∈ V
16 ceqsex8v.9 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
17163anbi3d 1213 . . . . 5 (𝑥 = 𝐴 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓)))
18174exbidv 1750 . . . 4 (𝑥 = 𝐴 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓)))
19 ceqsex8v.10 . . . . . 6 (𝑦 = 𝐵 → (𝜓𝜒))
20193anbi3d 1213 . . . . 5 (𝑦 = 𝐵 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒)))
21204exbidv 1750 . . . 4 (𝑦 = 𝐵 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒)))
22 ceqsex8v.11 . . . . . 6 (𝑧 = 𝐶 → (𝜒𝜃))
23223anbi3d 1213 . . . . 5 (𝑧 = 𝐶 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃)))
24234exbidv 1750 . . . 4 (𝑧 = 𝐶 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃)))
25 ceqsex8v.12 . . . . . 6 (𝑤 = 𝐷 → (𝜃𝜏))
26253anbi3d 1213 . . . . 5 (𝑤 = 𝐷 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏)))
27264exbidv 1750 . . . 4 (𝑤 = 𝐷 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏)))
2812, 13, 14, 15, 18, 21, 24, 27ceqsex4v 2597 . . 3 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏))
29 ceqsex8v.5 . . . 4 𝐸 ∈ V
30 ceqsex8v.6 . . . 4 𝐹 ∈ V
31 ceqsex8v.7 . . . 4 𝐺 ∈ V
32 ceqsex8v.8 . . . 4 𝐻 ∈ V
33 ceqsex8v.13 . . . 4 (𝑣 = 𝐸 → (𝜏𝜂))
34 ceqsex8v.14 . . . 4 (𝑢 = 𝐹 → (𝜂𝜁))
35 ceqsex8v.15 . . . 4 (𝑡 = 𝐺 → (𝜁𝜎))
36 ceqsex8v.16 . . . 4 (𝑠 = 𝐻 → (𝜎𝜌))
3729, 30, 31, 32, 33, 34, 35, 36ceqsex4v 2597 . . 3 (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏) ↔ 𝜌)
3828, 37bitri 173 . 2 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ 𝜌)
3911, 38bitri 173 1 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator