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Theorem ceqsex3v 2569
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 383 . . . . . 6 (((x = A (y = B z = 𝐶)) φ) ↔ (x = A ((y = B z = 𝐶) φ)))
2 3anass 875 . . . . . . 7 ((x = A y = B z = 𝐶) ↔ (x = A (y = B z = 𝐶)))
32anbi1i 434 . . . . . 6 (((x = A y = B z = 𝐶) φ) ↔ ((x = A (y = B z = 𝐶)) φ))
4 df-3an 873 . . . . . . 7 ((y = B z = 𝐶 φ) ↔ ((y = B z = 𝐶) φ))
54anbi2i 433 . . . . . 6 ((x = A (y = B z = 𝐶 φ)) ↔ (x = A ((y = B z = 𝐶) φ)))
61, 3, 53bitr4i 201 . . . . 5 (((x = A y = B z = 𝐶) φ) ↔ (x = A (y = B z = 𝐶 φ)))
762exbii 1475 . . . 4 (yz((x = A y = B z = 𝐶) φ) ↔ yz(x = A (y = B z = 𝐶 φ)))
8 19.42vv 1766 . . . 4 (yz(x = A (y = B z = 𝐶 φ)) ↔ (x = A yz(y = B z = 𝐶 φ)))
97, 8bitri 173 . . 3 (yz((x = A y = B z = 𝐶) φ) ↔ (x = A yz(y = B z = 𝐶 φ)))
109exbii 1474 . 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 1196 . . . . 5 (x = A → ((y = B z = 𝐶 φ) ↔ (y = B z = 𝐶 ψ)))
14132exbidv 1726 . . . 4 (x = A → (yz(y = B z = 𝐶 φ) ↔ yz(y = B z = 𝐶 ψ)))
1511, 14ceqsexv 2566 . . 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 2568 . . 3 (yz(y = B z = 𝐶 ψ) ↔ θ)
2115, 20bitri 173 . 2 (x(x = A yz(y = B z = 𝐶 φ)) ↔ θ)
2210, 21bitri 173 1 (xyz((x = A y = B z = 𝐶) φ) ↔ θ)
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:  ceqsex6v  2571
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