Theorem nfra2wOLD

Description: Obsolete version of nfra2w as of 31-Oct-2024. (Contributed by Alan Sare, 31-Dec-2011) (Revised by Gino Giotto, 24-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nfra2wOLD	⊢ 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