Theorem ee4anv

Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv . (Contributed by NM, 31-Jul-1995) Remove disjoint variable conditions on y , z and x , w . (Revised by Eric Schmidt, 26-Oct-2025)

	Ref	Expression
Assertion	ee4anv	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		excom	⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) )
2	1	exbii	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) )
3		eeanv	⊢ ( ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) )
4	3	2exbii	⊢ ( ∃ 𝑥 ∃ 𝑧 ∃ 𝑦 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ∃ 𝑥 ∃ 𝑧 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) )
5		nfv	⊢ Ⅎ 𝑧 𝜑
6	5	nfex	⊢ Ⅎ 𝑧 ∃ 𝑦 𝜑
7		nfv	⊢ Ⅎ 𝑥 𝜓
8	7	nfex	⊢ Ⅎ 𝑥 ∃ 𝑤 𝜓
9	6 8	eean	⊢ ( ∃ 𝑥 ∃ 𝑧 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )
10	2 4 9	3bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )