Theorem eean

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

	Ref	Expression
Hypotheses	eean.1	⊢ Ⅎ 𝑦 𝜑
	eean.2	⊢ Ⅎ 𝑥 𝜓
Assertion	eean	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		eean.1	⊢ Ⅎ 𝑦 𝜑
2		eean.2	⊢ Ⅎ 𝑥 𝜓
3	1	19.42	⊢ ( ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
4	3	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
5	2	nfex	⊢ Ⅎ 𝑥 ∃ 𝑦 𝜓
6	5	19.41	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )
7	4 6	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )