Theorem raaan 3469

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 3468	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
4	3	pm5.74i 179	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) ↔ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)))
5		ralm 3467	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓))
6		jcab 592	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) ↔ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ∧ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓)))
7		ralm 3467	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ ∀𝑥 ∈ 𝐴 𝜑)
8		eleq1 2202	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	8	cbvexv 1890	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
10	9	imbi1i 237	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ (∃𝑦 𝑦 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓))
11		ralm 3467	. . . . 5 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ ∀𝑦 ∈ 𝐴 𝜓)
12	10, 11	bitri 183	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓) ↔ ∀𝑦 ∈ 𝐴 𝜓)
13	7, 12	anbi12i 455	. . 3 ⊢ (((∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ∧ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
14	6, 13	bitri 183	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))
15	4, 5, 14	3bitr3i 209	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑦 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480  ∀wral 2416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421

This theorem is referenced by:  raaanv  3470