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Theorem ceqsex6v 2899
 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 938 . . . . 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 1584 . . . 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 1908 . . . 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 240 . . 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 1584 . 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 684 . . . . 5 (x = A → (((w = D v = E u = F) φ) ↔ ((w = D v = E u = F) ψ)))
11103exbidv 1629 . . . 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 684 . . . . 5 (y = B → (((w = D v = E u = F) ψ) ↔ ((w = D v = E u = F) χ)))
14133exbidv 1629 . . . 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 684 . . . . 5 (z = C → (((w = D v = E u = F) χ) ↔ ((w = D v = E u = F) θ)))
17163exbidv 1629 . . . 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 2897 . . 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 2897 . . 3 (wvu((w = D v = E u = F) θ) ↔ ζ)
2618, 25bitri 240 . 2 (xyz((x = A y = B z = C) wvu((w = D v = E u = F) φ)) ↔ ζ)
275, 26bitri 240 1 (xyzwvu((x = A y = B z = C) (w = D v = E u = F) φ) ↔ ζ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by: (None)
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