Theorem vtocl3 1380

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 group(s):   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 1352	. . . 4 ⊢ ∃x x = A
3		vtocl3.2	. . . . 5 ⊢ B ∈ V
4	3	isseti 1352	. . . 4 ⊢ ∃y y = B
5		vtocl3.3	. . . . 5 ⊢ C ∈ V
6	5	isseti 1352	. . . 4 ⊢ ∃z z = C
7		eeeanv 981	. . . . 5 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
8		vtocl3.4	. . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
9	8	biimpd 135	. . . . . . . 8 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ → ψ))
10	9	19.22i 723	. . . . . . 7 ⊢ (∃z(x = A ∧ y = B ∧ z = C) → ∃z(φ → ψ))
11	10	19.22i 723	. . . . . 6 ⊢ (∃y∃z(x = A ∧ y = B ∧ z = C) → ∃y∃z(φ → ψ))
12	11	19.22i 723	. . . . 5 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) → ∃x∃y∃z(φ → ψ))
13	7, 12	sylbir 176	. . . 4 ⊢ ((∃x x = A ∧ ∃y y = B ∧ ∃z z = C) → ∃x∃y∃z(φ → ψ))
14	2, 4, 6, 13	mp3an 642	. . 3 ⊢ ∃x∃y∃z(φ → ψ)
15		19.36v 958	. . . . . . 7 ⊢ (∃z(φ → ψ) ↔ (∀zφ → ψ))
16	15	biex 733	. . . . . 6 ⊢ (∃y∃z(φ → ψ) ↔ ∃y(∀zφ → ψ))
17		19.36v 958	. . . . . 6 ⊢ (∃y(∀zφ → ψ) ↔ (∀y∀zφ → ψ))
18	16, 17	bitr 151	. . . . 5 ⊢ (∃y∃z(φ → ψ) ↔ (∀y∀zφ → ψ))
19	18	biex 733	. . . 4 ⊢ (∃x∃y∃z(φ → ψ) ↔ ∃x(∀y∀zφ → ψ))
20		19.36v 958	. . . 4 ⊢ (∃x(∀y∀zφ → ψ) ↔ (∀x∀y∀zφ → ψ))
21	19, 20	bitr 151	. . 3 ⊢ (∃x∃y∃z(φ → ψ) ↔ (∀x∀y∀zφ → ψ))
22	14, 21	mpbi 164	. 2 ⊢ (∀x∀y∀zφ → ψ)
23		vtocl3.5	. . 3 ⊢ φ
24	23	gen2 681	. 2 ⊢ ∀y∀zφ
25	22, 24	mpg 684	1 ⊢ ψ

Syntax hints:   → wi 2   ↔ wb 127   ∧ w3a 581  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  caoprass 3068  caoprdistr 3073  ertr 3211

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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