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Metamath Proof Explorer

Theorem exdistrv

Description: Distribute a pair of existential quantifiers (over disjoint variables) over a conjunction. Combination of 19.41v and 19.42v . For a version with fewer disjoint variable conditions but requiring more axioms, see eeanv . (Contributed by BJ, 30-Sep-2022)

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

Proof

Step	Hyp	Ref	Expression
1		exdistr	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2		19.41v	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )
3	1 2	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )