Theorem setinds

Description: Principle of set induction (or _E -induction). If a property passes from all elements of x to x itself, then it holds for all x . (Contributed by Scott Fenton, 10-Mar-2011)

	Ref	Expression
Hypothesis	setinds.1	|- ( A. y e. x [. y / x ]. ph -> ph )
Assertion	setinds	|- ph

Proof

Step	Hyp	Ref	Expression
1		setinds.1	|- ( A. y e. x [. y / x ]. ph -> ph )
2		vex	|- x e. _V
3		setind	|- ( A. z ( z C_ { x | ph } -> z e. { x | ph } ) -> { x | ph } = _V )
4		dfss3	|- ( z C_ { x | ph } <-> A. y e. z y e. { x | ph } )
5		df-sbc	|- ( [. y / x ]. ph <-> y e. { x | ph } )
6	5	ralbii	|- ( A. y e. z [. y / x ]. ph <-> A. y e. z y e. { x | ph } )
7		nfcv	|- F/_ x z
8		nfsbc1v	|- F/ x [. y / x ]. ph
9	7 8	nfralw	|- F/ x A. y e. z [. y / x ]. ph
10		nfsbc1v	|- F/ x [. z / x ]. ph
11	9 10	nfim	|- F/ x ( A. y e. z [. y / x ]. ph -> [. z / x ]. ph )
12		raleq	|- ( x = z -> ( A. y e. x [. y / x ]. ph <-> A. y e. z [. y / x ]. ph ) )
13		sbceq1a	|- ( x = z -> ( ph <-> [. z / x ]. ph ) )
14	12 13	imbi12d	|- ( x = z -> ( ( A. y e. x [. y / x ]. ph -> ph ) <-> ( A. y e. z [. y / x ]. ph -> [. z / x ]. ph ) ) )
15	11 14 1	chvarfv	|- ( A. y e. z [. y / x ]. ph -> [. z / x ]. ph )
16	6 15	sylbir	|- ( A. y e. z y e. { x | ph } -> [. z / x ]. ph )
17	4 16	sylbi	|- ( z C_ { x | ph } -> [. z / x ]. ph )
18		df-sbc	|- ( [. z / x ]. ph <-> z e. { x | ph } )
19	17 18	sylib	|- ( z C_ { x | ph } -> z e. { x | ph } )
20	3 19	mpg	|- { x | ph } = _V
21	20	eqcomi	|- _V = { x | ph }
22	21	abeq2i	|- ( x e. _V <-> ph )
23	2 22	mpbi	|- ph