Theorem ralidmw 4443

Description: Idempotent law for restricted quantifier. Weak version of ralidm 4447, which does not require ax-10 2140, ax-12 2174, but requires ax-8 2111. (Contributed by Gino Giotto, 30-Sep-2024.)

Hypothesis

Ref	Expression
ralidmw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ralidmw	⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem ralidmw

Step	Hyp	Ref	Expression
1		df-ral 3070	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2	1	imbi2i 335	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3	2	albii 1825	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
4		pm2.21 123	. . . . . 6 ⊢ (¬ 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → 𝜑))
5		eleq1w 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6		ralidmw.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	5, 6	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
8	7	spw 2040	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))
9	4, 8	ja 186	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → (𝑥 ∈ 𝐴 → 𝜑))
10	9	alimi 1817	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
11	7	hba1w 2053	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
12		ax-1 6	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
13	11, 12	alrimih 1829	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
14	10, 13	impbii 208	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
15	3, 14	bitri 274	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
16		df-ral 3070	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
17	15, 16, 1	3bitr4i 302	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2109  ∀wral 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-clel 2817  df-ral 3070

This theorem is referenced by:  dfwe2  7615