Theorem vtocl3 1847

Description: Implicit substitution of classes for set variables.

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

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.1	. . . . 5 ⊢ A ∈ V
2	1	isseti 1818	. . . 4 ⊢ ∃x x = A
3		vtocl3.2	. . . . 5 ⊢ B ∈ V
4	3	isseti 1818	. . . 4 ⊢ ∃y y = B
5		vtocl3.3	. . . . 5 ⊢ C ∈ V
6	5	isseti 1818	. . . 4 ⊢ ∃z z = C
7		eeeanv 1326	. . . . 5 ⊢ (∃x∃y∃z(x = A ⋀ y = B ⋀ z = C) ↔ (∃x x = A ⋀ ∃y y = B ⋀ ∃z z = C))
8		vtocl3.4	. . . . . . . 8 ⊢ ((x = A ⋀ y = B ⋀ z = C) → (φ ↔ ψ))
9	8	biimpd 153	. . . . . . 7 ⊢ ((x = A ⋀ y = B ⋀ z = C) → (φ → ψ))
10	9	19.22i 1042	. . . . . 6 ⊢ (∃z(x = A ⋀ y = B ⋀ z = C) → ∃z(φ → ψ))
11	10	19.22i2 1043	. . . . 5 ⊢ (∃x∃y∃z(x = A ⋀ y = B ⋀ z = C) → ∃x∃y∃z(φ → ψ))
12	7, 11	sylbir 201	. . . 4 ⊢ ((∃x x = A ⋀ ∃y y = B ⋀ ∃z z = C) → ∃x∃y∃z(φ → ψ))
13	2, 4, 6, 12	mp3an 918	. . 3 ⊢ ∃x∃y∃z(φ → ψ)
14		19.36v 1302	. . . . . . 7 ⊢ (∃z(φ → ψ) ↔ (∀zφ → ψ))
15	14	exbii 1053	. . . . . 6 ⊢ (∃y∃z(φ → ψ) ↔ ∃y(∀zφ → ψ))
16		19.36v 1302	. . . . . 6 ⊢ (∃y(∀zφ → ψ) ↔ (∀y∀zφ → ψ))
17	15, 16	bitr 173	. . . . 5 ⊢ (∃y∃z(φ → ψ) ↔ (∀y∀zφ → ψ))
18	17	exbii 1053	. . . 4 ⊢ (∃x∃y∃z(φ → ψ) ↔ ∃x(∀y∀zφ → ψ))
19		19.36v 1302	. . . 4 ⊢ (∃x(∀y∀zφ → ψ) ↔ (∀x∀y∀zφ → ψ))
20	18, 19	bitr 173	. . 3 ⊢ (∃x∃y∃z(φ → ψ) ↔ (∀x∀y∀zφ → ψ))
21	13, 20	mpbi 189	. 2 ⊢ (∀x∀y∀zφ → ψ)
22		vtocl3.5	. . 3 ⊢ φ
23	22	gen2 985	. 2 ⊢ ∀y∀zφ
24	21, 23	mpg 988	1 ⊢ ψ

Syntax hints:   → wi 3   ↔ wb 146   ⋀ w3a 777  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814

This theorem is referenced by:  caoprass 4060  caoprdistr 4065  ertr 4280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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