Theorem raleqf 1321

Description: Equality theorem for restricted universal quantifier, with bound variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
raleqf.1	⊢ (y ∈ A → ∀x y ∈ A)
raleqf.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
raleqf	⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

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

Proof of Theorem raleqf

Step	Hyp	Ref	Expression
1		raleqf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		raleqf.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1171	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1150	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	imbi1d 465	. . 3 ⊢ (A = B → ((x ∈ A → φ) ↔ (x ∈ B → φ)))
6	3, 5	biald 782	. 2 ⊢ (A = B → (∀x(x ∈ A → φ) ↔ ∀x(x ∈ B → φ)))
7		df-ral 1205	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
8		df-ral 1205	. 2 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
9	6, 7, 8	3bitr4g 428	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  raleq 1324  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-9 799  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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