Theorem ssabral

Description: The relation for a subclass of a class abstraction is equivalent to restricted quantification. (Contributed by NM, 6-Sep-2006)

	Ref	Expression
Assertion	ssabral	⊢ ( 𝐴 ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ssab	⊢ ( 𝐴 ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3	1 2	bitr4i	⊢ ( 𝐴 ⊆ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )