Theorem raaan 1775

Description: Rearrange restricted quantifiers.

Assertion

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

Distinct variable group(s):   φ,y   ψ,x   x,y,A

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		pm5.1 501	. . 3 ⊢ ((∀x ∈ A ∀y ∈ A (φ ∧ ψ) ∧ (∀x ∈ A φ ∧ ∀y ∈ A ψ)) → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
2		rzal 1773	. . 3 ⊢ (A = ∅ → ∀x ∈ A ∀y ∈ A (φ ∧ ψ))
3		rzal 1773	. . . 4 ⊢ (A = ∅ → ∀x ∈ A φ)
4		rzal 1773	. . . 4 ⊢ (A = ∅ → ∀y ∈ A ψ)
5	3, 4	jca 236	. . 3 ⊢ (A = ∅ → (∀x ∈ A φ ∧ ∀y ∈ A ψ))
6	1, 2, 5	sylanc 361	. 2 ⊢ (A = ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
7		r19.3rzv 1767	. . . . . 6 ⊢ (¬ A = ∅ → (φ ↔ ∀y ∈ A φ))
8	7	anbi1d 469	. . . . 5 ⊢ (¬ A = ∅ → ((φ ∧ ∀y ∈ A ψ) ↔ (∀y ∈ A φ ∧ ∀y ∈ A ψ)))
9		r19.26 1289	. . . . 5 ⊢ (∀y ∈ A (φ ∧ ψ) ↔ (∀y ∈ A φ ∧ ∀y ∈ A ψ))
10	8, 9	syl6rbbr 417	. . . 4 ⊢ (¬ A = ∅ → (∀y ∈ A (φ ∧ ψ) ↔ (φ ∧ ∀y ∈ A ψ)))
11	10	biraldv 1219	. . 3 ⊢ (¬ A = ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ ∀x ∈ A (φ ∧ ∀y ∈ A ψ)))
12		r19.3rzv 1767	. . . . 5 ⊢ (¬ A = ∅ → (∀y ∈ A ψ ↔ ∀x ∈ A ∀y ∈ A ψ))
13	12	anbi2d 468	. . . 4 ⊢ (¬ A = ∅ → ((∀x ∈ A φ ∧ ∀y ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ψ)))
14		r19.26 1289	. . . 4 ⊢ (∀x ∈ A (φ ∧ ∀y ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ψ))
15	13, 14	syl6rbbr 417	. . 3 ⊢ (¬ A = ∅ → (∀x ∈ A (φ ∧ ∀y ∈ A ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
16	11, 15	bitrd 406	. 2 ⊢ (¬ A = ∅ → (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ)))
17	6, 16	pm2.61i 110	1 ⊢ (∀x ∈ A ∀y ∈ A (φ ∧ ψ) ↔ (∀x ∈ A φ ∧ ∀y ∈ A ψ))

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196   = wceq 1091  ∀wral 1201  ∅c0 1707

This theorem is referenced by:  hlimcaui 5141

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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