Theorem reeanv

Description: Rearrange restricted existential quantifiers. (Contributed by NM, 9-May-1999)

	Ref	Expression
Assertion	reeanv	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		exdistrv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) ↔ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ∧ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
2	1	reeanlem	⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑦 ∈ 𝐵 𝜓 ) )