Theorem eean 1931

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 1688	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
3	2	exbii 1605	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4		eean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 1637	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.41 1686	. 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 1460  ∃wex 1492

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  eeanv  1932  reean  2646