Theorem ralbi 2751

Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)

Assertion

Ref	Expression
ralbi	⊢ (∀x ∈ A (φ ↔ ψ) → (∀x ∈ A φ ↔ ∀x ∈ A ψ))

Proof of Theorem ralbi

Step	Hyp	Ref	Expression
1		nfra1 2665	. 2 ⊢ Ⅎx∀x ∈ A (φ ↔ ψ)
2		rsp 2675	. . 3 ⊢ (∀x ∈ A (φ ↔ ψ) → (x ∈ A → (φ ↔ ψ)))
3	2	imp 418	. 2 ⊢ ((∀x ∈ A (φ ↔ ψ) ∧ x ∈ A) → (φ ↔ ψ))
4	1, 3	ralbida 2629	1 ⊢ (∀x ∈ A (φ ↔ ψ) → (∀x ∈ A φ ↔ ∀x ∈ A ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  uniiunlem  3354  iineq2  3987