Theorem ralimdv2

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005)

	Ref	Expression
Hypothesis	ralimdv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) → ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
Assertion	ralimdv2	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐵 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ralimdv2.1	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 → 𝜓 ) → ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
2	1	alimdv	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜒 ) ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜒 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜒 ) )
5	2 3 4	3imtr4g	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → ∀ 𝑥 ∈ 𝐵 𝜒 ) )