Theorem rabid2

Description: An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	rabid2	$$⊢A=\left\{x\in A|\phi \right\}\leftrightarrow \forall x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		abeq2	$$⊢A=\left\{x|\left(x\in A\wedge \phi \right)\right\}\leftrightarrow \forall x\left(x\in A\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
2		pm4.71	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in A\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
3	2	albii	$$⊢\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall x\left(x\in A\leftrightarrow \left(x\in A\wedge \phi \right)\right)$$
4	1 3	bitr4i	$$⊢A=\left\{x|\left(x\in A\wedge \phi \right)\right\}\leftrightarrow \forall x\left(x\in A\to \phi \right)$$
5		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
6	5	eqeq2i	$$⊢A=\left\{x\in A|\phi \right\}\leftrightarrow A=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
7		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
8	4 6 7	3bitr4i	$$⊢A=\left\{x\in A|\phi \right\}\leftrightarrow \forall x\in A\phi$$