Theorem rsp2

Description: Restricted specialization, with two quantifiers. (Contributed by NM, 11-Feb-1997)

	Ref	Expression
Assertion	rsp2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		rsp	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝐵 𝜑 ) )
2		rsp	⊢ ( ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑦 ∈ 𝐵 → 𝜑 ) )
3	1 2	syl6	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → 𝜑 ) ) )
4	3	impd	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐵 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝜑 ) )