Theorem excom13

Description: Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995)

	Ref	Expression
Assertion	excom13	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		excom	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑦 ∃ 𝑥 ∃ 𝑧 𝜑 )
2		excom	⊢ ( ∃ 𝑥 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑥 𝜑 )
3	2	exbii	⊢ ( ∃ 𝑦 ∃ 𝑥 ∃ 𝑧 𝜑 ↔ ∃ 𝑦 ∃ 𝑧 ∃ 𝑥 𝜑 )
4		excom	⊢ ( ∃ 𝑦 ∃ 𝑧 ∃ 𝑥 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 )
5	1 3 4	3bitri	⊢ ( ∃ 𝑥 ∃ 𝑦 ∃ 𝑧 𝜑 ↔ ∃ 𝑧 ∃ 𝑦 ∃ 𝑥 𝜑 )