Theorem nfra2wOLD 3155

Description: Obsolete version of nfra2w 3154 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.)

Assertion

Ref	Expression
nfra2wOLD	⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem nfra2wOLD

Step	Hyp	Ref	Expression
1		df-ral 3069	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑))
2		df-ral 3069	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))
3	2	imbi2i 336	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
4	3	albii 1822	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
5	1, 4	bitri 274	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
6		nfa1 2148	. . 3 ⊢ Ⅎ𝑦∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))
7		alcom 2156	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ ∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)))
8		19.21v 1942	. . . . . 6 ⊢ (∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
9	8	albii 1822	. . . . 5 ⊢ (∀𝑥∀𝑦(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
10	7, 9	bitri 274	. . . 4 ⊢ (∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
11	10	nfbii 1854	. . 3 ⊢ (Ⅎ𝑦∀𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑)) ↔ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑)))
12	6, 11	mpbi 229	. 2 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜑))
13	5, 12	nfxfr 1855	1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑

Syntax hints:   → wi 4  ∀wal 1537  Ⅎwnf 1786   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-10 2137  ax-11 2154

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-nf 1787  df-ral 3069

This theorem is referenced by: (None)