Theorem ssralv

Description: Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006) Avoid axioms. (Revised by GG, 19-May-2025)

	Ref	Expression
Assertion	ssralv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2		imim1	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
3	2	al2imi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
4		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜑 ) )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6	3 4 5	3imtr4g	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
7	1 6	sylbi	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )