Theorem alrot4

Description: Rotate four universal quantifiers twice. (Contributed by NM, 2-Feb-2005) (Proof shortened by Fan Zheng, 6-Jun-2016)

	Ref	Expression
Assertion	alrot4	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 ∀ 𝑥 ∀ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		alrot3	⊢ ( ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 ∀ 𝑦 𝜑 )
2	1	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 𝜑 ↔ ∀ 𝑥 ∀ 𝑧 ∀ 𝑤 ∀ 𝑦 𝜑 )
3		alrot3	⊢ ( ∀ 𝑥 ∀ 𝑧 ∀ 𝑤 ∀ 𝑦 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 ∀ 𝑥 ∀ 𝑦 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 𝜑 ↔ ∀ 𝑧 ∀ 𝑤 ∀ 𝑥 ∀ 𝑦 𝜑 )