Theorem ralrot3

Description: Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021)

	Ref	Expression
Assertion	ralrot3	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ralcom	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 𝜑 )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 𝜑 )
3		ralcom	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 )