Theorem 19.42vvv

Description: Version of 19.42 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011) (Proof shortened by Wolf Lammen, 27-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		exdistr2	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 ∃ 𝑧 𝜓 ) )
2		19.42v	⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 ∃ 𝑧 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 ) )
3	1 2	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜓 ) )