Theorem setindis 13859

Description: Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)

Hypotheses

Ref	Expression
setindis.nf0	|- F/ x ps
setindis.nf1	|- F/ x ch
setindis.nf2	|- F/ y ph
setindis.nf3	|- F/ y ps
setindis.1	|- ( x = z -> ( ph -> ps ) )
setindis.2	|- ( x = y -> ( ch -> ph ) )

Assertion

Ref	Expression
setindis	|- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )

Distinct variable groups:   x, y, z   ph, z

Allowed substitution hints:   ph( x, y)   ps( x, y, z)   ch( x, y, z)

Proof of Theorem setindis

Step	Hyp	Ref	Expression
1		nfcv 2308	. . . . 5 |- F/_ x y
2		setindis.nf0	. . . . 5 |- F/ x ps
3	1, 2	nfralxy 2504	. . . 4 |- F/ x A. z e. y ps
4		setindis.nf1	. . . 4 |- F/ x ch
5	3, 4	nfim 1560	. . 3 |- F/ x ( A. z e. y ps -> ch )
6		nfcv 2308	. . . . 5 |- F/_ y x
7		setindis.nf3	. . . . 5 |- F/ y ps
8	6, 7	nfralxy 2504	. . . 4 |- F/ y A. z e. x ps
9		setindis.nf2	. . . 4 |- F/ y ph
10	8, 9	nfim 1560	. . 3 |- F/ y ( A. z e. x ps -> ph )
11		raleq 2661	. . . . 5 |- ( y = x -> ( A. z e. y ps <-> A. z e. x ps ) )
12	11	biimprd 157	. . . 4 |- ( y = x -> ( A. z e. x ps -> A. z e. y ps ) )
13		setindis.2	. . . . 5 |- ( x = y -> ( ch -> ph ) )
14	13	equcoms 1696	. . . 4 |- ( y = x -> ( ch -> ph ) )
15	12, 14	imim12d 74	. . 3 |- ( y = x -> ( ( A. z e. y ps -> ch ) -> ( A. z e. x ps -> ph ) ) )
16	5, 10, 15	cbv3 1730	. 2 |- ( A. y ( A. z e. y ps -> ch ) -> A. x ( A. z e. x ps -> ph ) )
17		setindis.1	. . . . . 6 |- ( x = z -> ( ph -> ps ) )
18	2, 17	bj-sbime 13664	. . . . 5 |- ( [ z / x ] ph -> ps )
19	18	ralimi 2529	. . . 4 |- ( A. z e. x [ z / x ] ph -> A. z e. x ps )
20	19	imim1i 60	. . 3 |- ( ( A. z e. x ps -> ph ) -> ( A. z e. x [ z / x ] ph -> ph ) )
21	20	alimi 1443	. 2 |- ( A. x ( A. z e. x ps -> ph ) -> A. x ( A. z e. x [ z / x ] ph -> ph ) )
22		ax-setind 4514	. 2 |- ( A. x ( A. z e. x [ z / x ] ph -> ph ) -> A. x ph )
23	16, 21, 22	3syl 17	1 |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1341   F/wnf 1448   [wsb 1750   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  bj-inf2vnlem4  13865  bj-findis  13871