Theorem rexcomf 2472

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 253	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
2	1	anbi1i 431	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
3	2	2exbii 1497	. . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
4		excom 1554	. . 3 ⊢ (∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
5	3, 4	bitri 173	. 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
6		ralcomf.1	. . 3 ⊢ Ⅎ𝑦𝐴
7	6	r2exf 2342	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
8		ralcomf.2	. . 3 ⊢ Ⅎ𝑥𝐵
9	8	r2exf 2342	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
10	5, 7, 9	3bitr4i 201	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312

This theorem is referenced by:  rexcom  2474