Theorem ralanid

Description: Cancellation law for restricted universal quantification. (Contributed by Peter Mazsa, 30-Dec-2018) (Proof shortened by Wolf Lammen, 29-Jun-2023)

	Ref	Expression
Assertion	ralanid	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ibar	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2	1	bicomd	⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ 𝜑 ) )
3	2	ralbiia	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )