Theorem rgen

Description: Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994)

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

Proof

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