Theorem ralss

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015) Avoid axioms. (Revised by SN, 14-Oct-2025)

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

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2		id	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
3	2	pm4.71rd	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ) )
4	3	imbi1d	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝜑 ) ) )
5		impexp	⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
6	4 5	bitrdi	⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
7	6	alimi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) → ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
8	1 7	sylbi	⊢ ( 𝐴 ⊆ 𝐵 → ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
9		albi	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
10	8 9	syl	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
11		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
12		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
13	10 11 12	3bitr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )