Theorem rexcom 1313

Description: Commutation of restricted quantifiers.

Assertion

Ref	Expression
rexcom	⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)

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

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		ancom 333	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) ↔ (y ∈ B ∧ x ∈ A))
2	1	anbi1i 368	. . . 4 ⊢ (((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ((y ∈ B ∧ x ∈ A) ∧ φ))
3	2	bi2ex 734	. . 3 ⊢ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ) ↔ ∃x∃y((y ∈ B ∧ x ∈ A) ∧ φ))
4		excom 728	. . 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		r2ex 1241	. 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ φ))
7		r2ex 1241	. 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:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  rexcom4 1361  creui 4533  pjthu2 5242  shscomt 5284  mdsymlem4 5779  mdsymlem8 5783

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-or 197  df-an 198  df-ex 679  df-rex 1206

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