Theorem setindis 16562

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 2374	. . . . 5 |- F/_ x y
2		setindis.nf0	. . . . 5 |- F/ x ps
3	1, 2	nfralxy 2570	. . . 4 |- F/ x A. z e. y ps
4		setindis.nf1	. . . 4 |- F/ x ch
5	3, 4	nfim 1620	. . 3 |- F/ x ( A. z e. y ps -> ch )
6		nfcv 2374	. . . . 5 |- F/_ y x
7		setindis.nf3	. . . . 5 |- F/ y ps
8	6, 7	nfralxy 2570	. . . 4 |- F/ y A. z e. x ps
9		setindis.nf2	. . . 4 |- F/ y ph
10	8, 9	nfim 1620	. . 3 |- F/ y ( A. z e. x ps -> ph )
11		raleq 2730	. . . . 5 |- ( y = x -> ( A. z e. y ps <-> A. z e. x ps ) )
12	11	biimprd 158	. . . 4 |- ( y = x -> ( A. z e. x ps -> A. z e. y ps ) )
13		setindis.2	. . . . 5 |- ( x = y -> ( ch -> ph ) )
14	13	equcoms 1756	. . . 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 1790	. 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 16369	. . . . 5 |- ( [ z / x ] ph -> ps )
19	18	ralimi 2595	. . . 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 1503	. 2 |- ( A. x ( A. z e. x ps -> ph ) -> A. x ( A. z e. x [ z / x ] ph -> ph ) )
22		ax-setind 4635	. 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 1395   F/wnf 1508   [wsb 1810   A.wral 2510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515

This theorem is referenced by:  bj-inf2vnlem4  16568  bj-findis  16574