Theorem ralidm 3382

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 2409	. . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑
2		anidm 388	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
3		rsp2 2425	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑))
4	2, 3	syl5bir 151	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 2444	. 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 2409	. . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
8	7	19.23 1613	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
9	6, 8	sylibr 132	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
10		df-ral 2364	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
11	9, 10	sylibr 132	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
12	5, 11	impbii 124	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1287  ∃wex 1426   ∈ wcel 1438  ∀wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-ral 2364

This theorem is referenced by:  issref  4814  cnvpom  4973