Theorem raaan 3347

Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Hypotheses

Ref	Expression
raaan.1	⊢ Ⅎ𝑦𝜑
raaan.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
raaan	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		raaan.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		raaan.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3	1, 2	raaanlem 3346	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
4	3	pm5.74i 178	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) ↔ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
5		ralm 3345	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
6		jcab 567	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) ↔ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ∧ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓)))
7		ralm 3345	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
8		eleq1 2141	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	cbvexv 1836	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
10	9	imbi1i 236	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ (∃𝑦 𝑦 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓))
11		ralm 3345	. . . . 5 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ ∀𝑦 ∈ 𝐴 𝜓)
12	10, 11	bitri 182	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ ∀𝑦 ∈ 𝐴 𝜓)
13	7, 12	anbi12i 447	. . 3 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ∧ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
14	6, 13	bitri 182	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
15	4, 5, 14	3bitr3i 208	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  Ⅎwnf 1389  ∃wex 1421   ∈ wcel 1433  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by:  raaanv  3348