Theorem ralidm 3654

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		rzal 3652	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀x ∈ A φ)
2		rzal 3652	. . 3 ⊢ (A = ∅ → ∀x ∈ A φ)
3	1, 2	2thd 231	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
4		neq0 3561	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
5		biimt 325	. . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ)))
6		df-ral 2620	. . . . 5 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x(x ∈ A → ∀x ∈ A φ))
7		nfra1 2665	. . . . . 6 ⊢ Ⅎx∀x ∈ A φ
8	7	19.23 1801	. . . . 5 ⊢ (∀x(x ∈ A → ∀x ∈ A φ) ↔ (∃x x ∈ A → ∀x ∈ A φ))
9	6, 8	bitri 240	. . . 4 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ (∃x x ∈ A → ∀x ∈ A φ))
10	5, 9	syl6rbbr 255	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
11	4, 10	sylbi 187	. 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ))
12	3, 11	pm2.61i 156	1 ⊢ (∀x ∈ A ∀x ∈ A φ ↔ ∀x ∈ A φ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)