Theorem vtocl3 3478

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 3426	. . . . . 6 ⊢ ∃𝑥 𝑥 = 𝐴
3		vtocl3.2	. . . . . . 7 ⊢ 𝐵 ∈ V
4	3	isseti 3426	. . . . . 6 ⊢ ∃𝑦 𝑦 = 𝐵
5		vtocl3.3	. . . . . . 7 ⊢ 𝐶 ∈ V
6	5	isseti 3426	. . . . . 6 ⊢ ∃𝑧 𝑧 = 𝐶
7		eeeanv 2374	. . . . . . 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 221	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 → 𝜓))
10	9	eximi 1933	. . . . . . . 8 ⊢ (∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑧(𝜑 → 𝜓))
11	10	2eximi 1934	. . . . . . 7 ⊢ (∃𝑥∃𝑦∃𝑧(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓))
12	7, 11	sylbir 227	. . . . . 6 ⊢ ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓))
13	2, 4, 6, 12	mp3an 1589	. . . . 5 ⊢ ∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓)
14		19.36v 2091	. . . . . 6 ⊢ (∃𝑧(𝜑 → 𝜓) ↔ (∀𝑧𝜑 → 𝜓))
15	14	2exbii 1948	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 → 𝜓) ↔ ∃𝑥∃𝑦(∀𝑧𝜑 → 𝜓))
16	13, 15	mpbi 222	. . . 4 ⊢ ∃𝑥∃𝑦(∀𝑧𝜑 → 𝜓)
17		19.36v 2091	. . . . 5 ⊢ (∃𝑦(∀𝑧𝜑 → 𝜓) ↔ (∀𝑦∀𝑧𝜑 → 𝜓))
18	17	exbii 1947	. . . 4 ⊢ (∃𝑥∃𝑦(∀𝑧𝜑 → 𝜓) ↔ ∃𝑥(∀𝑦∀𝑧𝜑 → 𝜓))
19	16, 18	mpbi 222	. . 3 ⊢ ∃𝑥(∀𝑦∀𝑧𝜑 → 𝜓)
20	19	19.36iv 2045	. 2 ⊢ (∀𝑥∀𝑦∀𝑧𝜑 → 𝜓)
21		vtocl3.5	. . 3 ⊢ 𝜑
22	21	gen2 1895	. 2 ⊢ ∀𝑦∀𝑧𝜑
23	20, 22	mpg 1896	1 ⊢ 𝜓

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1111  ∀wal 1654   = wceq 1656  ∃wex 1878   ∈ wcel 2164  Vcvv 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416

This theorem is referenced by: (None)