Theorem spc3gv 2823

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 2744	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2744	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3		elisset 2744	. . . 4 ⊢ (𝐶 ∈ 𝑋 → ∃𝑧 𝑧 = 𝐶)
4	1, 2, 3	3anim123i 1179	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5		eeeanv 1926	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
6	4, 5	sylibr 133	. 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 158	. . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜓))
9	8	2alimi 1449	. . . . . 6 ⊢ (∀𝑦∀𝑧𝜑 → ∀𝑦∀𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜓))
10	9	alimi 1448	. . . . 5 ⊢ (∀𝑥∀𝑦∀𝑧𝜑 → ∀𝑥∀𝑦∀𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜓))
11		exim 1592	. . . . . 6 ⊢ (∀𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜓) → (∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧𝜓))
12	11	2alimi 1449	. . . . 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 1592	. . . . 5 ⊢ (∀𝑦(∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧𝜓) → (∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑦∃𝑧𝜓))
15	14	alimi 1448	. . . 4 ⊢ (∀𝑥∀𝑦(∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧𝜓) → ∀𝑥(∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑦∃𝑧𝜓))
16		exim 1592	. . . 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 1864	. . . 4 ⊢ (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑦∃𝑧𝜓)
19		19.9v 1864	. . . 4 ⊢ (∃𝑦∃𝑧𝜓 ↔ ∃𝑧𝜓)
20		19.9v 1864	. . . 4 ⊢ (∃𝑧𝜓 ↔ 𝜓)
21	18, 19, 20	3bitri 205	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧𝜓 ↔ 𝜓)
22	17, 21	syl6ib 160	. 2 ⊢ (∀𝑥∀𝑦∀𝑧𝜑 → (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → 𝜓))
23	6, 22	syl5com 29	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 973  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  funopg  5232