Theorem ee4anvOLD

Description: Obsolete version of ee4anv as of 26-Oct-2025. (Contributed by NM, 31-Jul-1995) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ee4anvOLD	⊢ ( ∃ 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		eeanv	⊢ ( ∃ 𝑥 ∃ 𝑧 ( ∃ 𝑦 𝜑 ∧ ∃ 𝑤 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )
6	2 4 5	3bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 ∃ 𝑤 ( 𝜑 ∧ 𝜓 ) ↔ ( ∃ 𝑥 ∃ 𝑦 𝜑 ∧ ∃ 𝑧 ∃ 𝑤 𝜓 ) )