Theorem bdsetindis 10764

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

Hypotheses

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

Assertion

Ref	Expression
bdsetindis	|- ( 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 bdsetindis

Step	Hyp	Ref	Expression
1		nfcv 2219	. . . . 5 |- F/_ x y
2		bdsetindis.nf0	. . . . 5 |- F/ x ps
3	1, 2	nfralxy 2402	. . . 4 |- F/ x A. z e. y ps
4		bdsetindis.nf1	. . . 4 |- F/ x ch
5	3, 4	nfim 1504	. . 3 |- F/ x ( A. z e. y ps -> ch )
6		nfcv 2219	. . . . 5 |- F/_ y x
7		bdsetindis.nf3	. . . . 5 |- F/ y ps
8	6, 7	nfralxy 2402	. . . 4 |- F/ y A. z e. x ps
9		bdsetindis.nf2	. . . 4 |- F/ y ph
10	8, 9	nfim 1504	. . 3 |- F/ y ( A. z e. x ps -> ph )
11		raleq 2549	. . . . 5 |- ( y = x -> ( A. z e. y ps <-> A. z e. x ps ) )
12	11	biimprd 156	. . . 4 |- ( y = x -> ( A. z e. x ps -> A. z e. y ps ) )
13		bdsetindis.2	. . . . 5 |- ( x = y -> ( ch -> ph ) )
14	13	equcoms 1634	. . . 4 |- ( y = x -> ( ch -> ph ) )
15	12, 14	imim12d 73	. . 3 |- ( y = x -> ( ( A. z e. y ps -> ch ) -> ( A. z e. x ps -> ph ) ) )
16	5, 10, 15	cbv3 1670	. 2 |- ( A. y ( A. z e. y ps -> ch ) -> A. x ( A. z e. x ps -> ph ) )
17		bdsetindis.1	. . . . . 6 |- ( x = z -> ( ph -> ps ) )
18	2, 17	bj-sbime 10584	. . . . 5 |- ( [ z / x ] ph -> ps )
19	18	ralimi 2426	. . . 4 |- ( A. z e. x [ z / x ] ph -> A. z e. x ps )
20	19	imim1i 59	. . 3 |- ( ( A. z e. x ps -> ph ) -> ( A. z e. x [ z / x ] ph -> ph ) )
21	20	alimi 1384	. 2 |- ( A. x ( A. z e. x ps -> ph ) -> A. x ( A. z e. x [ z / x ] ph -> ph ) )
22		bdsetindis.bd	. . 3 |- BOUNDED ph
23	22	ax-bdsetind 10763	. 2 |- ( A. x ( A. z e. x [ z / x ] ph -> ph ) -> A. x ph )
24	16, 21, 23	3syl 17	1 |- ( A. y ( A. z e. y ps -> ch ) -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1282   F/wnf 1389   [wsb 1685   A.wral 2348  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bdsetind 10763

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by:  bj-inf2vnlem3  10767