Theorem r19.15 1292

Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification.

Assertion

Ref	Expression
r19.15	⊢ (∀x ∈ A (φ ↔ ψ) → (∀x ∈ A φ ↔ ∀x ∈ A ψ))

Proof of Theorem r19.15

Step	Hyp	Ref	Expression
1		hbra1 1237	. . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → ∀x∀x ∈ A (φ ↔ ψ))
2		id 9	. . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → ∀x ∈ A (φ ↔ ψ))
3	1, 2	19.21ai 740	. 2 ⊢ (∀x ∈ A (φ ↔ ψ) → ∀x∀x ∈ A (φ ↔ ψ))
4		ra4 1243	. . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → (x ∈ A → (φ ↔ ψ)))
5	4	imp 277	. 2 ⊢ ((∀x ∈ A (φ ↔ ψ) ∧ x ∈ A) → (φ ↔ ψ))
6	3, 5	biralda 1213	1 ⊢ (∀x ∈ A (φ ↔ ψ) → (∀x ∈ A φ ↔ ∀x ∈ A ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  isowe 2941  rankonid 3538  kmlem12 3591  kmlem13 3592

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-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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