Theorem spc3gv 2639

Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Hypothesis

Ref	Expression
spc3egv.1	⊢ ((x = A ∧ y = B ∧ z = 𝐶) → (φ ↔ ψ))

Assertion

Ref	Expression
spc3gv	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀x∀y∀zφ → ψ))

Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   ψ,x,y,z

Allowed substitution hints:   φ(x,y,z)   𝑉(x,y,z)   𝑊(x,y,z)   𝑋(x,y,z)

Proof of Theorem spc3gv

Step	Hyp	Ref	Expression
1		elisset 2562	. . . 4 ⊢ (A ∈ 𝑉 → ∃x x = A)
2		elisset 2562	. . . 4 ⊢ (B ∈ 𝑊 → ∃y y = B)
3		elisset 2562	. . . 4 ⊢ (𝐶 ∈ 𝑋 → ∃z z = 𝐶)
4	1, 2, 3	3anim123i 1088	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃x x = A ∧ ∃y y = B ∧ ∃z z = 𝐶))
5		eeeanv 1805	. . 3 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = 𝐶) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = 𝐶))
6	4, 5	sylibr 137	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ∃x∃y∃z(x = A ∧ y = B ∧ z = 𝐶))
7		spc3egv.1	. . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = 𝐶) → (φ ↔ ψ))
8	7	biimpcd 148	. . . . . . 7 ⊢ (φ → ((x = A ∧ y = B ∧ z = 𝐶) → ψ))
9	8	2alimi 1342	. . . . . 6 ⊢ (∀y∀zφ → ∀y∀z((x = A ∧ y = B ∧ z = 𝐶) → ψ))
10	9	alimi 1341	. . . . 5 ⊢ (∀x∀y∀zφ → ∀x∀y∀z((x = A ∧ y = B ∧ z = 𝐶) → ψ))
11		exim 1487	. . . . . 6 ⊢ (∀z((x = A ∧ y = B ∧ z = 𝐶) → ψ) → (∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃zψ))
12	11	2alimi 1342	. . . . 5 ⊢ (∀x∀y∀z((x = A ∧ y = B ∧ z = 𝐶) → ψ) → ∀x∀y(∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃zψ))
13	10, 12	syl 14	. . . 4 ⊢ (∀x∀y∀zφ → ∀x∀y(∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃zψ))
14		exim 1487	. . . . 5 ⊢ (∀y(∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃zψ) → (∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃y∃zψ))
15	14	alimi 1341	. . . 4 ⊢ (∀x∀y(∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃zψ) → ∀x(∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃y∃zψ))
16		exim 1487	. . . 4 ⊢ (∀x(∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃y∃zψ) → (∃x∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃x∃y∃zψ))
17	13, 15, 16	3syl 17	. . 3 ⊢ (∀x∀y∀zφ → (∃x∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ∃x∃y∃zψ))
18		19.9v 1748	. . . 4 ⊢ (∃x∃y∃zψ ↔ ∃y∃zψ)
19		19.9v 1748	. . . 4 ⊢ (∃y∃zψ ↔ ∃zψ)
20		19.9v 1748	. . . 4 ⊢ (∃zψ ↔ ψ)
21	18, 19, 20	3bitri 195	. . 3 ⊢ (∃x∃y∃zψ ↔ ψ)
22	17, 21	syl6ib 150	. 2 ⊢ (∀x∀y∀zφ → (∃x∃y∃z(x = A ∧ y = B ∧ z = 𝐶) → ψ))
23	6, 22	syl5com 26	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀x∀y∀zφ → ψ))

Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 884  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390

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:  funopg  4877