Theorem eean 1984

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 1736	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
3	2	exbii 1653	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4		eean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 1685	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.41 1734	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
7	3, 6	bitri 184	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  Ⅎwnf 1508  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509

This theorem is referenced by:  eeanv  1985  reean  2702