Theorem rabss

Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006)

	Ref	Expression
Assertion	rabss	$$⊢\left\{x\in A|\phi \right\}\subseteq B\leftrightarrow \forall x\in A\left(\phi \to x\in B\right)$$

Proof

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