Theorem expandral

Description: Expand a restricted universal quantifier to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypothesis	expandral.1	⊢ ( 𝜑 ↔ 𝜓 )
Assertion	expandral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		expandral.1	⊢ ( 𝜑 ↔ 𝜓 )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜓 )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4	2 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )