Theorem ralcomf 2770

Description: Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
ralcomf.1	⊢ ℲyA
ralcomf.2	⊢ ℲxB

Assertion

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

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)   A(x,y)   B(x,y)

Proof of Theorem ralcomf

Step	Hyp	Ref	Expression
1		ancomsimp 1369	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) → φ) ↔ ((y ∈ B ∧ x ∈ A) → φ))
2	1	2albii 1567	. . 3 ⊢ (∀x∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀x∀y((y ∈ B ∧ x ∈ A) → φ))
3		alcom 1737	. . 3 ⊢ (∀x∀y((y ∈ B ∧ x ∈ A) → φ) ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
4	2, 3	bitri 240	. 2 ⊢ (∀x∀y((x ∈ A ∧ y ∈ B) → φ) ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
5		ralcomf.1	. . 3 ⊢ ℲyA
6	5	r2alf 2650	. 2 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀x∀y((x ∈ A ∧ y ∈ B) → φ))
7		ralcomf.2	. . 3 ⊢ ℲxB
8	7	r2alf 2650	. 2 ⊢ (∀y ∈ B ∀x ∈ A φ ↔ ∀y∀x((y ∈ B ∧ x ∈ A) → φ))
9	4, 6, 8	3bitr4i 268	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  Ⅎwnfc 2477  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  ralcom  2772  ssiinf  4016