Theorem 3ralbii

Description: Inference adding three restricted universal quantifiers to both sides of an equivalence. (Contributed by Peter Mazsa, 25-Jul-2019)

	Ref	Expression
Hypothesis	3ralbii.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	3ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐶 𝜓 )

Proof

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