Theorem raleq1f 1786

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

Hypotheses

Ref	Expression
raleq1f.1	⊢ (y ∈ A → ∀x y ∈ A)
raleq1f.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
raleq1f	⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem raleq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		raleq1f.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1568	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1538	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	imbi1d 615	. . 3 ⊢ (A = B → ((x ∈ A → φ) ↔ (x ∈ B → φ)))
6	3, 5	albid 1106	. 2 ⊢ (A = B → (∀x(x ∈ A → φ) ↔ ∀x(x ∈ B → φ)))
7		df-ral 1652	. 2 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
8		df-ral 1652	. 2 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
9	6, 7, 8	3bitr4g 557	1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 956   = wceq 958   ∈ wcel 960  ∀wral 1648

This theorem is referenced by:  raleq1 1789  hta 4738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475  df-ral 1652

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