Theorem ralidm 2347

Description: Idempotent law for restricted quantifier.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem ralidm

Step	Hyp	Ref	Expression
1		pm5.1 674	. . 3 ⊢ ((∀x ∈ A ∀x ∈ A φ ⋀ ∀x ∈ A φ) → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
2		rzal 2345	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀x ∈ A φ)
3		rzal 2345	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
4	1, 2, 3	sylanc 471	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
5		n0 2279	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
6		biimt 729	. . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ)))
7		df-ral 1641	. . . . 5 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x(x ∈ A → ∀x ∈ A φ))
8		hbra1 1679	. . . . . 6 ⊢ (∀x ∈ A φ → ∀x∀x ∈ A φ)
9	8	19.23 1059	. . . . 5 ⊢ (∀x(x ∈ A → ∀x ∈ A φ) ↔ (∃x x ∈ A → ∀x ∈ A φ))
10	7, 9	bitr 173	. . . 4 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ))
11	6, 10	syl6rbbr 537	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
12	5, 11	sylbi 199	. 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
13	4, 12	pm2.61i 126	1 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146  ∀wal 951   = wceq 953   ∈ wcel 955  ∃wex 977  ∀wral 1637  ∅c0 2270

This theorem is referenced by:  dfwe2 2925  cnvpo 3508

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-nul 2271

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