Theorem ralrid

Description: Sufficient condition for the restricted universal quantifier. Deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		ralrid.1	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
3	1 2	sylibr	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )