Theorem ralimi2

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004)

	Ref	Expression
Hypothesis	ralimi2.1	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐵 → 𝜓 ) )
Assertion	ralimi2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralimi2.1	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐵 → 𝜓 ) )
2	1	alimi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) )
5	2 3 4	3imtr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )