Theorem ralcom13

Description: Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021)

	Ref	Expression
Assertion	ralcom13	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ralcom	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝜑 )
2		ralcom	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 𝜑 )
3	2	ralbii	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 𝜑 )
4		ralcom	⊢ ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 𝜑 )
5	1 3 4	3bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑧 ∈ 𝐶 ∀ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐴 𝜑 )