Theorem clelab

Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993) (Proof shortened by Wolf Lammen, 16-Nov-2019) Avoid ax-11 , see sbc5ALT for more details. (Revised by SN, 2-Sep-2024)

	Ref	Expression
Assertion	clelab	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=A\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		elissetv	$$⊢A\in \left\{x|\phi \right\}\to \exists yy=A$$
2		exsimpl	$$⊢\exists x\left(x=A\wedge \phi \right)\to \exists xx=A$$
3		eqeq1	$$⊢x=y\to \left(x=A\leftrightarrow y=A\right)$$
4	3	cbvexvw	$$⊢\exists xx=A\leftrightarrow \exists yy=A$$
5	2 4	sylib	$$⊢\exists x\left(x=A\wedge \phi \right)\to \exists yy=A$$
6		eleq1	$$⊢y=A\to \left(y\in \left\{x|\phi \right\}\leftrightarrow A\in \left\{x|\phi \right\}\right)$$
7		df-clab	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \left[y/x\right]\phi$$
8		sb5	$$⊢\left[y/x\right]\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$
9	7 8	bitri	$$⊢y\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=y\wedge \phi \right)$$
10		eqeq2	$$⊢y=A\to \left(x=y\leftrightarrow x=A\right)$$
11	10	anbi1d	$$⊢y=A\to \left(\left(x=y\wedge \phi \right)\leftrightarrow \left(x=A\wedge \phi \right)\right)$$
12	11	exbidv	$$⊢y=A\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
13	9 12	bitrid	$$⊢y=A\to \left(y\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
14	6 13	bitr3d	$$⊢y=A\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
15	14	exlimiv	$$⊢\exists yy=A\to \left(A\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=A\wedge \phi \right)\right)$$
16	1 5 15	pm5.21nii	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \exists x\left(x=A\wedge \phi \right)$$