Theorem ssabf

Description: Subclass of a class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	ssabf.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	ssabf	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ssabf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2	1	abid2f	$$⊢\left\{x|x\in A\right\}=A$$
3	2	sseq1i	$$⊢\left\{x|x\in A\right\}\subseteq \left\{x|\phi \right\}\leftrightarrow A\subseteq \left\{x|\phi \right\}$$
4		ss2ab	$$⊢\left\{x|x\in A\right\}\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$
5	3 4	bitr3i	$$⊢A\subseteq \left\{x|\phi \right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$