Theorem rabssf

Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021)

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

Proof

Step	Hyp	Ref	Expression
1		rabssf.1	$$⊢\underset{\_}{Ⅎ}xB$$
2		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
3	2	sseq1i	$$⊢\left\{x\in A|\phi \right\}\subseteq B\leftrightarrow \left\{x|\left(x\in A\wedge \phi \right)\right\}\subseteq B$$
4	1	abssf	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}\subseteq B\leftrightarrow \forall x\left(\left(x\in A\wedge \phi \right)\to x\in B\right)$$
5		impexp	$$⊢\left(\left(x\in A\wedge \phi \right)\to x\in B\right)\leftrightarrow \left(x\in A\to \left(\phi \to x\in B\right)\right)$$
6	5	albii	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to x\in B\right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to x\in B\right)\right)$$
7		df-ral	$$⊢\forall x\in A\left(\phi \to x\in B\right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to x\in B\right)\right)$$
8	6 7	bitr4i	$$⊢\forall x\left(\left(x\in A\wedge \phi \right)\to x\in B\right)\leftrightarrow \forall x\in A\left(\phi \to x\in B\right)$$
9	3 4 8	3bitri	$$⊢\left\{x\in A|\phi \right\}\subseteq B\leftrightarrow \forall x\in A\left(\phi \to x\in B\right)$$