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Theorem ceqsex6v 2592
Description: Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
Hypotheses
Ref Expression
ceqsex6v.1 A V
ceqsex6v.2 B V
ceqsex6v.3 𝐶 V
ceqsex6v.4 𝐷 V
ceqsex6v.5 𝐸 V
ceqsex6v.6 𝐹 V
ceqsex6v.7 (x = A → (φψ))
ceqsex6v.8 (y = B → (ψχ))
ceqsex6v.9 (z = 𝐶 → (χθ))
ceqsex6v.10 (w = 𝐷 → (θτ))
ceqsex6v.11 (v = 𝐸 → (τη))
ceqsex6v.12 (u = 𝐹 → (ηζ))
Assertion
Ref Expression
ceqsex6v (xyzwvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ζ)
Distinct variable groups:   x,y,z,w,v,u,A   x,B,y,z,w,v,u   x,𝐶,y,z,w,v,u   x,𝐷,y,z,w,v,u   x,𝐸,y,z,w,v,u   x,𝐹,y,z,w,v,u   ψ,x   χ,y   θ,z   τ,w   η,v   ζ,u
Allowed substitution hints:   φ(x,y,z,w,v,u)   ψ(y,z,w,v,u)   χ(x,z,w,v,u)   θ(x,y,w,v,u)   τ(x,y,z,v,u)   η(x,y,z,w,u)   ζ(x,y,z,w,v)

Proof of Theorem ceqsex6v
StepHypRef Expression
1 3anass 888 . . . . 5 (((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ((x = A y = B z = 𝐶) ((w = 𝐷 v = 𝐸 u = 𝐹) φ)))
213exbii 1495 . . . 4 (wvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ wvu((x = A y = B z = 𝐶) ((w = 𝐷 v = 𝐸 u = 𝐹) φ)))
3 19.42vvv 1786 . . . 4 (wvu((x = A y = B z = 𝐶) ((w = 𝐷 v = 𝐸 u = 𝐹) φ)) ↔ ((x = A y = B z = 𝐶) wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ)))
42, 3bitri 173 . . 3 (wvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ((x = A y = B z = 𝐶) wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ)))
543exbii 1495 . 2 (xyzwvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ xyz((x = A y = B z = 𝐶) wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ)))
6 ceqsex6v.1 . . . 4 A V
7 ceqsex6v.2 . . . 4 B V
8 ceqsex6v.3 . . . 4 𝐶 V
9 ceqsex6v.7 . . . . . 6 (x = A → (φψ))
109anbi2d 437 . . . . 5 (x = A → (((w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ((w = 𝐷 v = 𝐸 u = 𝐹) ψ)))
11103exbidv 1746 . . . 4 (x = A → (wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ wvu((w = 𝐷 v = 𝐸 u = 𝐹) ψ)))
12 ceqsex6v.8 . . . . . 6 (y = B → (ψχ))
1312anbi2d 437 . . . . 5 (y = B → (((w = 𝐷 v = 𝐸 u = 𝐹) ψ) ↔ ((w = 𝐷 v = 𝐸 u = 𝐹) χ)))
14133exbidv 1746 . . . 4 (y = B → (wvu((w = 𝐷 v = 𝐸 u = 𝐹) ψ) ↔ wvu((w = 𝐷 v = 𝐸 u = 𝐹) χ)))
15 ceqsex6v.9 . . . . . 6 (z = 𝐶 → (χθ))
1615anbi2d 437 . . . . 5 (z = 𝐶 → (((w = 𝐷 v = 𝐸 u = 𝐹) χ) ↔ ((w = 𝐷 v = 𝐸 u = 𝐹) θ)))
17163exbidv 1746 . . . 4 (z = 𝐶 → (wvu((w = 𝐷 v = 𝐸 u = 𝐹) χ) ↔ wvu((w = 𝐷 v = 𝐸 u = 𝐹) θ)))
186, 7, 8, 11, 14, 17ceqsex3v 2590 . . 3 (xyz((x = A y = B z = 𝐶) wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ)) ↔ wvu((w = 𝐷 v = 𝐸 u = 𝐹) θ))
19 ceqsex6v.4 . . . 4 𝐷 V
20 ceqsex6v.5 . . . 4 𝐸 V
21 ceqsex6v.6 . . . 4 𝐹 V
22 ceqsex6v.10 . . . 4 (w = 𝐷 → (θτ))
23 ceqsex6v.11 . . . 4 (v = 𝐸 → (τη))
24 ceqsex6v.12 . . . 4 (u = 𝐹 → (ηζ))
2519, 20, 21, 22, 23, 24ceqsex3v 2590 . . 3 (wvu((w = 𝐷 v = 𝐸 u = 𝐹) θ) ↔ ζ)
2618, 25bitri 173 . 2 (xyz((x = A y = B z = 𝐶) wvu((w = 𝐷 v = 𝐸 u = 𝐹) φ)) ↔ ζ)
275, 26bitri 173 1 (xyzwvu((x = A y = B z = 𝐶) (w = 𝐷 v = 𝐸 u = 𝐹) φ) ↔ ζ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551
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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by: (None)
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