Theorem ralidm 3321

Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralidm

Step	Hyp	Ref	Expression
1		nfra1 2355	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑
2		anidm 376	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
3		rsp2 2371	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑))
4	2, 3	syl5bir 142	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 2390	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
6		ax-1 5	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
7		nfra1 2355	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
8	7	19.23 1568	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
9	6, 8	sylibr 137	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
10		df-ral 2311	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
11	9, 10	sylibr 137	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
12	5, 11	impbii 117	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  issref  4707  cnvpom  4860