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Metamath Proof Explorer

Theorem eeanv

Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v and 19.42v . For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv . (Contributed by NM, 26-Jul-1995)

		Ref	Expression
	Assertion	eeanv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	1 2	eean	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )