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	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		ssab	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$
2		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
3	1 2	bitr4i	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\in A\phi$$