Theorem ralidm 1774

Description: Idempotent law for restricted quantifier.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem ralidm

Step	Hyp	Ref	Expression
1		pm5.1 501	. . 3 ⊢ ((∀x ∈ A ∀x ∈ A φ ∧ ∀x ∈ A φ) → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
2		rzal 1773	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀x ∈ A φ)
3		rzal 1773	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
4	1, 2, 3	sylanc 361	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
5		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
6		biimt 549	. . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ)))
7		df-ral 1205	. . . . 5 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x(x ∈ A → ∀x ∈ A φ))
8		hbra1 1237	. . . . . 6 ⊢ (∀x ∈ A φ → ∀x∀x ∈ A φ)
9	8	19.23 745	. . . . 5 ⊢ (∀x(x ∈ A → ∀x ∈ A φ) ↔ (∃x x ∈ A → ∀x ∈ A φ))
10	7, 9	bitr 151	. . . 4 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ))
11	6, 10	syl6rbbr 417	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
12	5, 11	sylbi 174	. 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
13	4, 12	pm2.61i 110	1 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∅c0 1707

This theorem is referenced by:  dfwe2 2187

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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