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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 GG, 24-Sep-2024) (Proof shortened by Wolf Lammen, 3-Jan-2025)

Ref Expression
Assertion nfra2w 
|- F/ y A. x e. A A. y e. B ph

Proof

Step Hyp Ref Expression
1 r2al 
 |-  ( A. x e. A A. y e. B ph <-> A. x A. y ( ( x e. A /\ y e. B ) -> ph ) )
2 nfa2 
 |-  F/ y A. x A. y ( ( x e. A /\ y e. B ) -> ph )
3 1 2 nfxfr 
 |-  F/ y A. x e. A A. y e. B ph