Theorem eean 1806

Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
eean.1	⊢ Ⅎ𝑦𝜑
eean.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
eean	⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem eean

Step	Hyp	Ref	Expression
1		eean.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	19.42 1578	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
3	2	exbii 1496	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4		eean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 1528	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.41 1576	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
7	3, 6	bitri 173	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 97   ↔ wb 98  Ⅎwnf 1349  ∃wex 1381

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  eeanv  1807  reean  2478