Theorem 19.41vvv

Description: Version of 19.41 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995)

	Ref	Expression
Assertion	19.41vvv	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		19.41vv	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ( ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) )
3		19.41v	⊢ ( ∃ 𝑥 ( ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) )
4	2 3	bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ∧ 𝜓 ) )