Theorem ralimdaa

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of Margaris p. 90. (Contributed by NM, 22-Sep-2003) (Proof shortened by Wolf Lammen, 29-Dec-2019)

	Ref	Expression
Hypotheses	ralimdaa.1	⊢ Ⅎ 𝑥 𝜑
	ralimdaa.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
Assertion	ralimdaa	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralimdaa.1	⊢ Ⅎ 𝑥 𝜑
2		ralimdaa.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 → 𝜒 ) )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
4	1 3	ralrimi	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
5		ralim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )
6	4 5	syl	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )