Theorem eean 2341

Description: Distribute 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 2224	. . 3 ⊢ (∃𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
3	2	exbii 1846	. 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4		eean.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfex 2313	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜓
6	5	19.41 2223	. 2 ⊢ (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
7	3, 6	bitri 274	1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2132  ax-11 2149  ax-12 2166

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1778  df-nf 1782

This theorem is referenced by:  eeanv  2342  reean  3299