Theorem vtocl2 1379

Description: Implicit substitution of classes for set variables.

Hypotheses

Ref	Expression
vtocl2.1	⊢ A ∈ V
vtocl2.2	⊢ B ∈ V
vtocl2.3	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
vtocl2.4	⊢ φ

Assertion

Ref	Expression
vtocl2	⊢ ψ

Distinct variable group(s):   x,y,A   x,B,y   ψ,x,y

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . 5 ⊢ A ∈ V
2	1	isseti 1352	. . . 4 ⊢ ∃x x = A
3		vtocl2.2	. . . . 5 ⊢ B ∈ V
4	3	isseti 1352	. . . 4 ⊢ ∃y y = B
5		eeanv 980	. . . . 5 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
6		vtocl2.3	. . . . . . . 8 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	6	biimpd 135	. . . . . . 7 ⊢ ((x = A ∧ y = B) → (φ → ψ))
8	7	19.22i 723	. . . . . 6 ⊢ (∃y(x = A ∧ y = B) → ∃y(φ → ψ))
9	8	19.22i 723	. . . . 5 ⊢ (∃x∃y(x = A ∧ y = B) → ∃x∃y(φ → ψ))
10	5, 9	sylbir 176	. . . 4 ⊢ ((∃x x = A ∧ ∃y y = B) → ∃x∃y(φ → ψ))
11	2, 4, 10	mp2an 520	. . 3 ⊢ ∃x∃y(φ → ψ)
12		19.36v 958	. . . . 5 ⊢ (∃y(φ → ψ) ↔ (∀yφ → ψ))
13	12	biex 733	. . . 4 ⊢ (∃x∃y(φ → ψ) ↔ ∃x(∀yφ → ψ))
14		19.36v 958	. . . 4 ⊢ (∃x(∀yφ → ψ) ↔ (∀x∀yφ → ψ))
15	13, 14	bitr 151	. . 3 ⊢ (∃x∃y(φ → ψ) ↔ (∀x∀yφ → ψ))
16	11, 15	mpbi 164	. 2 ⊢ (∀x∀yφ → ψ)
17		vtocl2.4	. . 3 ⊢ φ
18	17	ax-gen 677	. 2 ⊢ ∀yφ
19	16, 18	mpg 684	1 ⊢ ψ

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  caoprcom 3067  caoprord 3070  ersym 3209

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-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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