Theorem rexcom 1822

Description: Commutation of restricted quantifiers.

Assertion

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

Distinct variable groups:   x,y   x,B   y,A

Proof of Theorem rexcom

Step	Hyp	Ref	Expression
1		ancom 446	. . . . 5 ⊢ ((x ∈ A ⋀ y ∈ B) ↔ (y ∈ B ⋀ x ∈ A))
2	1	anbi1i 492	. . . 4 ⊢ (((x ∈ A ⋀ y ∈ B) ⋀ φ) ↔ ((y ∈ B ⋀ x ∈ A) ⋀ φ))
3	2	2exbii 1093	. . 3 ⊢ (∃x∃y((x ∈ A ⋀ y ∈ B) ⋀ φ) ↔ ∃x∃y((y ∈ B ⋀ x ∈ A) ⋀ φ))
4		excom 1087	. . 3 ⊢ (∃x∃y((y ∈ B ⋀ x ∈ A) ⋀ φ) ↔ ∃y∃x((y ∈ B ⋀ x ∈ A) ⋀ φ))
5	3, 4	bitri 180	. 2 ⊢ (∃x∃y((x ∈ A ⋀ y ∈ B) ⋀ φ) ↔ ∃y∃x((y ∈ B ⋀ x ∈ A) ⋀ φ))
6		r2ex 1738	. 2 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃x∃y((x ∈ A ⋀ y ∈ B) ⋀ φ))
7		r2ex 1738	. 2 ⊢ (∃y ∈ B ∃x ∈ A φ ↔ ∃y∃x((y ∈ B ⋀ x ∈ A) ⋀ φ))
8	5, 6, 7	3bitr4i 190	1 ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)

Syntax hints:   ↔ wb 153   ⋀ wa 230   ∈ wcel 999  ∃wex 1021  ∃wrex 1693

This theorem is referenced by:  rexcom4 1871  brdom7disj 4866  creui 6833  shscom 9366  mdsymlem4 10417  mdsymlem8 10421

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-rex 1697

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