Theorem bdsetindis 14581

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 2319	. . . . 5 |- F/_ x y
2		bdsetindis.nf0	. . . . 5 |- F/ x ps
3	1, 2	nfralxy 2515	. . . 4 |- F/ x A. z e. y ps
4		bdsetindis.nf1	. . . 4 |- F/ x ch
5	3, 4	nfim 1572	. . 3 |- F/ x ( A. z e. y ps -> ch )
6		nfcv 2319	. . . . 5 |- F/_ y x
7		bdsetindis.nf3	. . . . 5 |- F/ y ps
8	6, 7	nfralxy 2515	. . . 4 |- F/ y A. z e. x ps
9		bdsetindis.nf2	. . . 4 |- F/ y ph
10	8, 9	nfim 1572	. . 3 |- F/ y ( A. z e. x ps -> ph )
11		raleq 2672	. . . . 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		bdsetindis.2	. . . . 5 |- ( x = y -> ( ch -> ph ) )
14	13	equcoms 1708	. . . 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 1742	. 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 14385	. . . . 5 |- ( [ z / x ] ph -> ps )
19	18	ralimi 2540	. . . 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 1455	. 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 14580	. 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 1351   F/wnf 1460   [wsb 1762   A.wral 2455  BOUNDED wbd 14424

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsetind 14580

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460

This theorem is referenced by:  bj-inf2vnlem3  14584