Theorem 3exdistr

Description: Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	3exdistr	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 ( 𝜓 ∧ ∃ 𝑧 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anass	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
2	1	2exbii	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) )
3		19.42vv	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜑 ∧ ∃ 𝑦 ∃ 𝑧 ( 𝜓 ∧ 𝜒 ) ) )
4		exdistr	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑦 ( 𝜓 ∧ ∃ 𝑧 𝜒 ) )
5	4	anbi2i	⊢ ( ( 𝜑 ∧ ∃ 𝑦 ∃ 𝑧 ( 𝜓 ∧ 𝜒 ) ) ↔ ( 𝜑 ∧ ∃ 𝑦 ( 𝜓 ∧ ∃ 𝑧 𝜒 ) ) )
6	2 3 5	3bitri	⊢ ( ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( 𝜑 ∧ ∃ 𝑦 ( 𝜓 ∧ ∃ 𝑧 𝜒 ) ) )
7	6	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜑 ∧ ∃ 𝑦 ( 𝜓 ∧ ∃ 𝑧 𝜒 ) ) )