Theorem rexcomf 2771

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

Hypotheses

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

Assertion

Ref	Expression
rexcomf	⊢ (∃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 rexcomf

Step	Hyp	Ref	Expression
1		ancom 437	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) ↔ (y ∈ B ∧ x ∈ A))
2	1	anbi1i 676	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ((y ∈ B ∧ x ∈ A) ∧ φ))
3	2	2exbii 1583	. . 3 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃x∃y((y ∈ B ∧ x ∈ A) ∧ φ))
4		excom 1741	. . 3 ⊢ (∃x∃y((y ∈ B ∧ x ∈ A) ∧ φ) ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ))
5	3, 4	bitri 240	. 2 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ))
6		ralcomf.1	. . 3 ⊢ ℲyA
7	6	r2exf 2651	. 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))
8		ralcomf.2	. . 3 ⊢ ℲxB
9	8	r2exf 2651	. 2 ⊢ (∃y ∈ B ∃x ∈ A φ ↔ ∃y∃x((y ∈ B ∧ x ∈ A) ∧ φ))
10	5, 7, 9	3bitr4i 268	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  Ⅎwnfc 2477  ∃wrex 2616

This proof depends on 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 proof 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-rex 2621

This theorem is used by:  rexcom  2773