Theorem eean 1912

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 1880	. . 3 ⊢ (∃y(φ ∧ ψ) ↔ (φ ∧ ∃yψ))
3	2	exbii 1582	. 2 ⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(φ ∧ ∃yψ))
4		eean.2	. . . 4 ⊢ Ⅎxψ
5	4	nfex 1843	. . 3 ⊢ Ⅎx∃yψ
6	5	19.41 1879	. 2 ⊢ (∃x(φ ∧ ∃yψ) ↔ (∃xφ ∧ ∃yψ))
7	3, 6	bitri 240	1 ⊢ (∃x∃y(φ ∧ ψ) ↔ (∃xφ ∧ ∃yψ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  eeanv  1913  reean  2778