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Theorem ceqsex3v 2897
 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 = A y = B z = 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 630 . . . . . 6 (((x = A (y = B z = C)) φ) ↔ (x = A ((y = B z = C) φ)))
2 3anass 938 . . . . . . 7 ((x = A y = B z = C) ↔ (x = A (y = B z = C)))
32anbi1i 676 . . . . . 6 (((x = A y = B z = C) φ) ↔ ((x = A (y = B z = C)) φ))
4 df-3an 936 . . . . . . 7 ((y = B z = C φ) ↔ ((y = B z = C) φ))
54anbi2i 675 . . . . . 6 ((x = A (y = B z = C φ)) ↔ (x = A ((y = B z = C) φ)))
61, 3, 53bitr4i 268 . . . . 5 (((x = A y = B z = C) φ) ↔ (x = A (y = B z = C φ)))
762exbii 1583 . . . 4 (yz((x = A y = B z = C) φ) ↔ yz(x = A (y = B z = C φ)))
8 19.42vv 1907 . . . 4 (yz(x = A (y = B z = C φ)) ↔ (x = A yz(y = B z = C φ)))
97, 8bitri 240 . . 3 (yz((x = A y = B z = C) φ) ↔ (x = A yz(y = B z = C φ)))
109exbii 1582 . 2 (xyz((x = A y = B z = C) φ) ↔ x(x = A yz(y = B z = C φ)))
11 ceqsex3v.1 . . . 4 A V
12 ceqsex3v.4 . . . . . 6 (x = A → (φψ))
13123anbi3d 1258 . . . . 5 (x = A → ((y = B z = C φ) ↔ (y = B z = C ψ)))
14132exbidv 1628 . . . 4 (x = A → (yz(y = B z = C φ) ↔ yz(y = B z = C ψ)))
1511, 14ceqsexv 2894 . . 3 (x(x = A yz(y = B z = C φ)) ↔ yz(y = B z = C ψ))
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 2896 . . 3 (yz(y = B z = C ψ) ↔ θ)
2115, 20bitri 240 . 2 (x(x = A yz(y = B z = C φ)) ↔ θ)
2210, 21bitri 240 1 (xyz((x = A y = B z = C) φ) ↔ θ)
 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  ceqsex6v  2899
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