Theorem rexcomf 2632

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 264	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
2	1	anbi1i 455	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
3	2	2exbii 1599	. . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
4		excom 1657	. . 3 ⊢ (∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
5	3, 4	bitri 183	. 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
6		ralcomf.1	. . 3 ⊢ Ⅎ𝑦𝐴
7	6	r2exf 2488	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
8		ralcomf.2	. . 3 ⊢ Ⅎ𝑥𝐵
9	8	r2exf 2488	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
10	5, 7, 9	3bitr4i 211	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1485   ∈ wcel 2141  Ⅎwnfc 2299  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  rexcom  2634