Theorem raaan 3307

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 3306	. . 3 ⊢ (∃x x ∈ A → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
4	3	pm5.74i 169	. 2 ⊢ ((∃x x ∈ A → ∀x ∈ A ∀y ∈ A (φ ∧ ψ)) ↔ (∃x x ∈ A → (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
5		ralm 3305	. 2 ⊢ ((∃x x ∈ A → ∀x ∈ A ∀y ∈ A (φ ∧ ψ)) ↔ ∀x ∈ A ∀y ∈ A (φ ∧ ψ))
6		jcab 522	. . 3 ⊢ ((∃x x ∈ A → (∀x ∈ A φ ∧ ∀y ∈ A ψ)) ↔ ((∃x x ∈ A → ∀x ∈ A φ) ∧ (∃x x ∈ A → ∀y ∈ A ψ)))
7		ralm 3305	. . . 4 ⊢ ((∃x x ∈ A → ∀x ∈ A φ) ↔ ∀x ∈ A φ)
8		eleq1 2082	. . . . . . 7 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
9	8	cbvexv 1777	. . . . . 6 ⊢ (∃x x ∈ A ↔ ∃y y ∈ A)
10	9	imbi1i 227	. . . . 5 ⊢ ((∃x x ∈ A → ∀y ∈ A ψ) ↔ (∃y y ∈ A → ∀y ∈ A ψ))
11		ralm 3305	. . . . 5 ⊢ ((∃y y ∈ A → ∀y ∈ A ψ) ↔ ∀y ∈ A ψ)
12	10, 11	bitri 173	. . . 4 ⊢ ((∃x x ∈ A → ∀y ∈ A ψ) ↔ ∀y ∈ A ψ)
13	7, 12	anbi12i 436	. . 3 ⊢ (((∃x x ∈ A → ∀x ∈ A φ) ∧ (∃x x ∈ A → ∀y ∈ A ψ)) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))
14	6, 13	bitri 173	. 2 ⊢ ((∃x x ∈ A → (∀x ∈ A φ ∧ ∀y ∈ A ψ)) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))
15	4, 5, 14	3bitr3i 199	1 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1328  ∃wex 1361   ∈ wcel 1375  ∀wral 2282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287

This theorem is referenced by:  raaanv  3308