Theorem 19.42vv

Description: Version of 19.42 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995)

	Ref	Expression
Assertion	19.42vv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		exdistr	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2		19.42v	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )
3	1 2	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 𝜓 ) )