Theorem ralrimd

Description: Inference from Theorem 19.21 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004)

	Ref	Expression
Hypotheses	ralrimd.1	⊢ Ⅎ 𝑥 𝜑
	ralrimd.2	⊢ Ⅎ 𝑥 𝜓
	ralrimd.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
Assertion	ralrimd	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralrimd.1	⊢ Ⅎ 𝑥 𝜑
2		ralrimd.2	⊢ Ⅎ 𝑥 𝜓
3		ralrimd.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
4	1 2 3	alrimd	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) ) )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜒 ) )
6	4 5	syl6ibr	⊢ ( 𝜑 → ( 𝜓 → ∀ 𝑥 ∈ 𝐴 𝜒 ) )