Theorem ralcom 1312

Description: Commutation of restricted quantifiers.

Assertion

Ref	Expression
ralcom	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

Distinct variable group(s):   x,y   x,B   y,A

Proof of Theorem ralcom

Step	Hyp	Ref	Expression
1		ancom 333	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) ↔ (y ∈ B ∧ x ∈ A))
2	1	imbi1i 161	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) → φ) ↔ ((y ∈ B ∧ x ∈ A) → φ))
3	2	bi2al 696	. . 3 ⊢ (∀x∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀x∀y((y ∈ B ∧ x ∈ A) → φ))
4		alcom 715	. . 3 ⊢ (∀x∀y((y ∈ B ∧ x ∈ A) → φ) ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
5	3, 4	bitr 151	. 2 ⊢ (∀x∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
6		r2al 1231	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → φ))
7		r2al 1231	. 2 ⊢ (∀y ∈ B ∀x ∈ A φ ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
8	5, 6, 7	3bitr4 158	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  ralcom4 1360  ssint 1980  fununi 2705  mapxpen 3390  occl 5188

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-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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