Theorem vtocl3 2828

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl3.1	⊢ 𝐴 ∈ V
vtocl3.2	⊢ 𝐵 ∈ V
vtocl3.3	⊢ 𝐶 ∈ V
vtocl3.4	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
vtocl3.5	⊢ 𝜑

Assertion

Ref	Expression
vtocl3	⊢ 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)

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.1	. . . . . . 7 ⊢ 𝐴 ∈ V
2	1	isseti 2779	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtocl3.2	. . . . . . 7 ⊢ 𝐵 ∈ V
4	3	isseti 2779	. . . . . 6 ⊢ ∃𝑦 𝑦 = 𝐵
5		vtocl3.3	. . . . . . 7 ⊢ 𝐶 ∈ V
6	5	isseti 2779	. . . . . 6 ⊢ ∃𝑧 𝑧 = 𝐶
7		eeeanv 1960	. . . . . . 7 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
8		vtocl3.4	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
9	8	biimpd 144	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 → 𝜓))
10	9	eximi 1622	. . . . . . . 8 ⊢ (∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧(𝜑 → 𝜓))
11	10	2eximi 1623	. . . . . . 7 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓))
12	7, 11	sylbir 135	. . . . . 6 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓))
13	2, 4, 6, 12	mp3an 1349	. . . . 5 ⊢ ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓)
14		nfv 1550	. . . . . . 7 ⊢ Ⅎ𝑧𝜓
15	14	19.36-1 1695	. . . . . 6 ⊢ (∃𝑧(𝜑 → 𝜓) → (∀𝑧𝜑 → 𝜓))
16	15	2eximi 1623	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓) → ∃𝑥∃𝑦(∀𝑧𝜑 → 𝜓))
17	13, 16	ax-mp 5	. . . 4 ⊢ ∃𝑥∃𝑦(∀𝑧𝜑 → 𝜓)
18		nfv 1550	. . . . 5 ⊢ Ⅎ𝑦𝜓
19	18	19.36-1 1695	. . . 4 ⊢ (∃𝑦(∀𝑧𝜑 → 𝜓) → (∀𝑦∀𝑧𝜑 → 𝜓))
20	17, 19	eximii 1624	. . 3 ⊢ ∃𝑥(∀𝑦∀𝑧𝜑 → 𝜓)
21	20	19.36aiv 1924	. 2 ⊢ (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓)
22		vtocl3.5	. . 3 ⊢ 𝜑
23	22	gen2 1472	. 2 ⊢ ∀𝑦∀𝑧𝜑
24	21, 23	mpg 1473	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 105   ∧ w3a 980  ∀wal 1370   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Vcvv 2771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-v 2773

This theorem is referenced by: (None)