Theorem ralss

Description: Restricted universal quantification on a subset in terms of superset. (Contributed by Stefan O'Rear, 3-Apr-2015)

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

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	pm4.71rd	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) ) )
3	2	imbi1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝜑 ) ) )
4		impexp	⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
5	3 4	bitrdi	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ) )
6	5	ralbidv2	⊢ ( 𝐴 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )