Metamath Proof Explorer

Theorem nfra2w

Description: Similar to Lemma 24 of Monk2 p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD . Version of nfra2 with a disjoint variable condition not requiring ax-13 . (Contributed by Alan Sare, 31-Dec-2011) (Revised by Gino Giotto, 24-Sep-2024) (Proof shortened by Wolf Lammen, 31-Oct-2024)

Ref Expression
Assertion nfra2w ⊢ Ⅎ y ∀ x ∈ A ∀ y ∈ B φ

Proof

Step Hyp Ref Expression
1 r2al ⊢ ∀ x ∈ A ∀ y ∈ B φ ↔ ∀ x ∀ y x ∈ A ∧ y ∈ B → φ
2 alcom ⊢ ∀ x ∀ y x ∈ A ∧ y ∈ B → φ ↔ ∀ y ∀ x x ∈ A ∧ y ∈ B → φ
3 1 2 bitri ⊢ ∀ x ∈ A ∀ y ∈ B φ ↔ ∀ y ∀ x x ∈ A ∧ y ∈ B → φ
4 nfa1 ⊢ Ⅎ y ∀ y ∀ x x ∈ A ∧ y ∈ B → φ
5 3 4 nfxfr ⊢ Ⅎ y ∀ x ∈ A ∀ y ∈ B φ