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, which does not require ax-13 . (Contributed by Alan Sare, 31-Dec-2011) (Revised by Gino Giotto, 24-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) )
2		df-ral	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) )
3	2	imbi2i	⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
4	3	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
5	1 4	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
6		nfa1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) )
7		alcom	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
8		19.21v	⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
9	8	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
10	7 9	bitri	⊢ ( ∀ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
11	10	nfbii	⊢ ( Ⅎ 𝑦 ∀ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) ↔ Ⅎ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
12	6 11	mpbi	⊢ Ⅎ 𝑦 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝐵 → 𝜑 ) )
13	5 12	nfxfr	⊢ Ⅎ 𝑦 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑