Theorem ralimiaa

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007)

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

Proof

Step	Hyp	Ref	Expression
1		ralimiaa.1	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 )
2	1	ex	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
3	2	ralimia	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )