Definition df-rex

Description: Define restricted existential quantification. Special case of Definition 4.15(4) of TakeutiZaring p. 22.

Note: This notation is most often used to express that ph holds for at least one element of a given class A . For this reading F/_ x A is required, though, for example, asserted when x and A are disjoint.

Should instead A depend on x , you rather assert at least one x fulfilling ph happens to be contained in the corresponding A ( x ) . This interpretation is rarely needed (see also df-ral ). (Contributed by NM, 30-Aug-1993)

	Ref	Expression
Assertion	df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		vx	$$setvarx$$
1		cA	$$classA$$
2		wph	$$wff\phi$$
3	2 0 1	wrex	$$wff\exists x\in A\phi$$
4	0	cv	$$setvarx$$
5	4 1	wcel	$$wffx\in A$$
6	5 2	wa	$$wff\left(x\in A\wedge \phi \right)$$
7	6 0	wex	$$wff\exists x\left(x\in A\wedge \phi \right)$$
8	3 7	wb	$$wff\left(\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)\right)$$