Theorem dfrab3

Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013)

	Ref	Expression
Assertion	dfrab3	$$⊢\left\{x\in A|\phi \right\}=A\cap \left\{x|\phi \right\}$$

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		inab	$$⊢\left\{x|x\in A\right\}\cap \left\{x|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
3		abid2	$$⊢\left\{x|x\in A\right\}=A$$
4	3	ineq1i	$$⊢\left\{x|x\in A\right\}\cap \left\{x|\phi \right\}=A\cap \left\{x|\phi \right\}$$
5	1 2 4	3eqtr2i	$$⊢\left\{x\in A|\phi \right\}=A\cap \left\{x|\phi \right\}$$