Theorem ralidmOLD 4443

Description: Obsolete version of ralidm 4439 as of 2-Sep-2024. (Contributed by NM, 28-Mar-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralidmOLD

Step	Hyp	Ref	Expression
1		rzal 4436	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑)
2		rzal 4436	. . 3 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
3	1, 2	2thd 264	. 2 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
4		neq0 4276	. . 3 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
5		df-ral 3068	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
6		nfra1 3142	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
7	6	19.23 2207	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
8	5, 7	bitri 274	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑))
9		biimt 360	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)))
10	8, 9	bitr4id 289	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
11	4, 10	sylbi 216	. 2 ⊢ (¬ 𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
12	3, 11	pm2.61i 182	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-ral 3068  df-dif 3886  df-nul 4254

This theorem is referenced by: (None)