Theorem eeor

Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994)

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

Proof

Step	Hyp	Ref	Expression
1		eeor.1	⊢ Ⅎ 𝑦 𝜑
2		eeor.2	⊢ Ⅎ 𝑥 𝜓
3	1	19.45	⊢ ( ∃ 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	19.44	⊢ ( ∃ 𝑥 ( 𝜑 ∨ ∃ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑦 𝜓 ) )
7	4 6	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∨ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∨ ∃ 𝑦 𝜓 ) )