Theorem vtocl2 1840

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 groups:   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 1812	. . . 4 ⊢ ∃x x = A
3		vtocl2.2	. . . . 5 ⊢ B ∈ V
4	3	isseti 1812	. . . 4 ⊢ ∃y y = B
5		eeanv 1322	. . . . 5 ⊢ (∃x∃y(x = A ⋀ y = B) ↔ (∃x x = A ⋀ ∃y y = B))
6		vtocl2.3	. . . . . . 7 ⊢ ((x = A ⋀ y = B) → (φ ↔ ψ))
7	6	biimpd 153	. . . . . 6 ⊢ ((x = A ⋀ y = B) → (φ → ψ))
8	7	19.22i2 1040	. . . . 5 ⊢ (∃x∃y(x = A ⋀ y = B) → ∃x∃y(φ → ψ))
9	5, 8	sylbir 201	. . . 4 ⊢ ((∃x x = A ⋀ ∃y y = B) → ∃x∃y(φ → ψ))
10	2, 4, 9	mp2an 696	. . 3 ⊢ ∃x∃y(φ → ψ)
11		19.36v 1299	. . . . 5 ⊢ (∃y(φ → ψ) ↔ (∀yφ → ψ))
12	11	exbii 1050	. . . 4 ⊢ (∃x∃y(φ → ψ) ↔ ∃x(∀yφ → ψ))
13		19.36v 1299	. . . 4 ⊢ (∃x(∀yφ → ψ) ↔ (∀x∀yφ → ψ))
14	12, 13	bitr 173	. . 3 ⊢ (∃x∃y(φ → ψ) ↔ (∀x∀yφ → ψ))
15	10, 14	mpbi 189	. 2 ⊢ (∀x∀yφ → ψ)
16		vtocl2.4	. . 3 ⊢ φ
17	16	ax-gen 962	. 2 ⊢ ∀yφ
18	15, 17	mpg 985	1 ⊢ ψ

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  ∃wex 979  Vcvv 1808

This theorem is referenced by:  caoprcom 4048  caoprord 4051  ersym 4265

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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