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Metamath Proof Explorer

Theorem nfralw

Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 1-Sep-1999) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 13-Dec-2024)

Ref Expression
Hypotheses nfralw.1 ⊢ Ⅎ _ x A
nfralw.2 ⊢ Ⅎ x φ
Assertion nfralw ⊢ Ⅎ x ∀ y ∈ A φ

Proof

Step Hyp Ref Expression
1 nfralw.1 ⊢ Ⅎ _ x A
2 nfralw.2 ⊢ Ⅎ x φ
3 1 nfcri ⊢ Ⅎ x y ∈ A
4 3 nf5ri ⊢ y ∈ A → ∀ x y ∈ A
5 2 nf5ri ⊢ φ → ∀ x φ
6 4 5 hbral ⊢ ∀ y ∈ A φ → ∀ x ∀ y ∈ A φ
7 6 nf5i ⊢ Ⅎ x ∀ y ∈ A φ