Theorem ralidm 2409

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		pm5.1 688	. . 3 ⊢ ((∀x ∈ A ∀x ∈ A φ ⋀ ∀x ∈ A φ) → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
2		rzal 2407	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀x ∈ A φ)
3		rzal 2407	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
4	1, 2, 3	sylanc 482	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
5		n0 2341	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
6		biimt 743	. . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ)))
7		df-ral 1696	. . . . 5 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x(x ∈ A → ∀x ∈ A φ))
8		hbra1 1734	. . . . . 6 ⊢ (∀x ∈ A φ → ∀x∀x ∈ A φ)
9	8	19.23 1104	. . . . 5 ⊢ (∀x(x ∈ A → ∀x ∈ A φ) ↔ (∃x x ∈ A → ∀x ∈ A φ))
10	7, 9	bitri 180	. . . 4 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ))
11	6, 10	syl6rbbr 550	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
12	5, 11	sylbi 206	. 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
13	4, 12	pm2.61i 132	1 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 153  ∀wal 995   = wceq 997   ∈ wcel 999  ∃wex 1021  ∀wral 1692  ∅c0 2331

This theorem is referenced by:  dfwe2 2992  cnvpo 3579  ref3w 10566

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-v 1859  df-dif 2100  df-nul 2332

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