Metamath Proof Explorer

Theorem aaan

Description: Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993)

Ref Expression
Hypotheses aaan.1 ⊢ Ⅎ y φ
aaan.2 ⊢ Ⅎ x ψ
Assertion aaan ⊢ ∀ x ∀ y φ ∧ ψ ↔ ∀ x φ ∧ ∀ y ψ

Proof

Step Hyp Ref Expression
1 aaan.1 ⊢ Ⅎ y φ
2 aaan.2 ⊢ Ⅎ x ψ
3 1 19.28 ⊢ ∀ y φ ∧ ψ ↔ φ ∧ ∀ y ψ
4 3 albii ⊢ ∀ x ∀ y φ ∧ ψ ↔ ∀ x φ ∧ ∀ y ψ
5 2 nfal ⊢ Ⅎ x ∀ y ψ
6 5 19.27 ⊢ ∀ x φ ∧ ∀ y ψ ↔ ∀ x φ ∧ ∀ y ψ
7 4 6 bitri ⊢ ∀ x ∀ y φ ∧ ψ ↔ ∀ x φ ∧ ∀ y ψ