Theorem cbvralf 1330

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvralf.1	⊢ (z ∈ A → ∀x z ∈ A)
cbvralf.2	⊢ (z ∈ A → ∀y z ∈ A)
cbvralf.3	⊢ (φ → ∀yφ)
cbvralf.4	⊢ (ψ → ∀xψ)
cbvralf.5	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvralf	⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Distinct variable group(s):   z,A   x,y,z

Proof of Theorem cbvralf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (z ∈ x → ∀y z ∈ x)
2		cbvralf.2	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
3	1, 2	hbel 1172	. . . 4 ⊢ (x ∈ A → ∀y x ∈ A)
4		cbvralf.3	. . . 4 ⊢ (φ → ∀yφ)
5	3, 4	hbim 702	. . 3 ⊢ ((x ∈ A → φ) → ∀y(x ∈ A → φ))
6		ax-17 925	. . . . 5 ⊢ (z ∈ y → ∀x z ∈ y)
7		cbvralf.1	. . . . 5 ⊢ (z ∈ A → ∀x z ∈ A)
8	6, 7	hbel 1172	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
9		cbvralf.4	. . . 4 ⊢ (ψ → ∀xψ)
10	8, 9	hbim 702	. . 3 ⊢ ((y ∈ A → ψ) → ∀x(y ∈ A → ψ))
11		eleq1 1149	. . . 4 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
12		cbvralf.5	. . . 4 ⊢ (x = y → (φ ↔ ψ))
13	11, 12	imbi12d 474	. . 3 ⊢ (x = y → ((x ∈ A → φ) ↔ (y ∈ A → ψ)))
14	5, 10, 13	cbval 848	. 2 ⊢ (∀x(x ∈ A → φ) ↔ ∀y(y ∈ A → ψ))
15		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
16		df-ral 1205	. 2 ⊢ (∀y ∈ A ψ ↔ ∀y(y ∈ A → ψ))
17	14, 15, 16	3bitr4 158	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797   ∈ wel 803   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  cbvral 1331  hta 3619

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-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205

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