Theorem eeeanv

Description: Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995) (Proof shortened by Andrew Salmon, 25-May-2011) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018)

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

Proof

Step	Hyp	Ref	Expression
1		eeanv	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) )
2	1	anbi1i	⊢ ( ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) ↔ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
3		df-3an	⊢ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
4	3	exbii	⊢ ( ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑧 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) )
5		19.42v	⊢ ( ∃ 𝑧 ( ( 𝜑 ∧ 𝜓 ) ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
6	4 5	bitri	⊢ ( ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
7	6	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
8		nfv	⊢ Ⅎ 𝑦 𝜒
9	8	nfex	⊢ Ⅎ 𝑦 ∃ 𝑧 𝜒
10	9	19.41	⊢ ( ∃ 𝑦 ( ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) ↔ ( ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
11	10	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ( ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) ↔ ∃ 𝑥 ( ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
12		nfv	⊢ Ⅎ 𝑥 𝜒
13	12	nfex	⊢ Ⅎ 𝑥 ∃ 𝑧 𝜒
14	13	19.41	⊢ ( ∃ 𝑥 ( ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
15	7 11 14	3bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ∃ 𝑥 ∃ 𝑦 ( 𝜑 ∧ 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
16		df-3an	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ∧ ∃ 𝑧 𝜒 ) ↔ ( ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ) ∧ ∃ 𝑧 𝜒 ) )
17	2 15 16	3bitr4i	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) ↔ ( ∃ 𝑥 𝜑 ∧ ∃ 𝑦 𝜓 ∧ ∃ 𝑧 𝜒 ) )