Theorem rexcomf 3277

Description: Commutation of restricted existential quantifiers. For a version based on fewer axioms see rexcom 3266. (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 460	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴))
2	1	anbi1i 624	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
3	2	2exbii 1849	. . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
4		excom 2163	. . 3 ⊢ (∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
5	3, 4	bitri 275	. 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
6		ralcomf.1	. . 3 ⊢ Ⅎ𝑦𝐴
7	6	r2exf 3259	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
8		ralcomf.2	. . 3 ⊢ Ⅎ𝑥𝐵
9	8	r2exf 3259	. 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑))
10	5, 7, 9	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  Ⅎwnfc 2876  ∃wrex 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054

This theorem is referenced by:  rexcom4f  32397