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Theorem ceqsex8v 2572
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 = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 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 1768 . . . . 5 (vu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
2 3anass 875 . . . . . . . 8 ((((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ)))
3 df-3an 873 . . . . . . . . 9 (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ) ↔ (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ))
43anbi2i 433 . . . . . . . 8 ((((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ)))
52, 4bitr4i 176 . . . . . . 7 ((((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
652exbii 1475 . . . . . 6 (𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ 𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
762exbii 1475 . . . . 5 (vu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ vu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
8 df-3an 873 . . . . 5 (((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)) ↔ (((x = A y = B) (z = 𝐶 w = 𝐷)) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
91, 7, 83bitr4i 201 . . . 4 (vu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ ((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
1092exbii 1475 . . 3 (zwvu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ zw((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)))
11102exbii 1475 . 2 (xyzwvu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ xyzw((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 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 1196 . . . . 5 (x = A → (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ) ↔ ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) ψ)))
18174exbidv 1728 . . . 4 (x = A → (vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ) ↔ vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) ψ)))
19 ceqsex8v.10 . . . . . 6 (y = B → (ψχ))
20193anbi3d 1196 . . . . 5 (y = B → (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) ψ) ↔ ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) χ)))
21204exbidv 1728 . . . 4 (y = B → (vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) ψ) ↔ vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) χ)))
22 ceqsex8v.11 . . . . . 6 (z = 𝐶 → (χθ))
23223anbi3d 1196 . . . . 5 (z = 𝐶 → (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) χ) ↔ ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) θ)))
24234exbidv 1728 . . . 4 (z = 𝐶 → (vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) χ) ↔ vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) θ)))
25 ceqsex8v.12 . . . . . 6 (w = 𝐷 → (θτ))
26253anbi3d 1196 . . . . 5 (w = 𝐷 → (((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) θ) ↔ ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) τ)))
27264exbidv 1728 . . . 4 (w = 𝐷 → (vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) θ) ↔ vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) τ)))
2812, 13, 14, 15, 18, 21, 24, 27ceqsex4v 2570 . . 3 (xyzw((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)) ↔ vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 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 2570 . . 3 (vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) τ) ↔ ρ)
3828, 37bitri 173 . 2 (xyzw((x = A y = B) (z = 𝐶 w = 𝐷) vu𝑡𝑠((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻) φ)) ↔ ρ)
3911, 38bitri 173 1 (xyzwvu𝑡𝑠(((x = A y = B) (z = 𝐶 w = 𝐷)) ((v = 𝐸 u = 𝐹) (𝑡 = 𝐺 𝑠 = 𝐻)) φ) ↔ ρ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 871   = wceq 1226  wex 1358   wcel 1370  Vcvv 2531
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-3an 873  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-v 2533
This theorem is referenced by: (None)
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