Metamath Proof Explorer


Theorem nfra2w

Description: Similar to Lemma 24 of Monk2 p. 114, except that quantification is restricted. Once derived from hbra2VD . Version of nfra2 with a disjoint variable condition not requiring ax-13 . (Contributed by Alan Sare, 31-Dec-2011) Reduce axiom usage. (Revised by Gino Giotto, 24-Sep-2024) (Proof shortened by Wolf Lammen, 3-Jan-2025)

Ref Expression
Assertion nfra2w 𝑦𝑥𝐴𝑦𝐵 𝜑

Proof

Step Hyp Ref Expression
1 r2al ( ∀ 𝑥𝐴𝑦𝐵 𝜑 ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → 𝜑 ) )
2 nfa2 𝑦𝑥𝑦 ( ( 𝑥𝐴𝑦𝐵 ) → 𝜑 )
3 1 2 nfxfr 𝑦𝑥𝐴𝑦𝐵 𝜑