Theorem bj-bdfindes 11727

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 11725 for explanations. From this version, it is easy to prove the bounded version of findes 4416. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-bdfindes.bd	|- BOUNDED ph

Assertion

Ref	Expression
bj-bdfindes	|- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Proof of Theorem bj-bdfindes

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1466	. . . 4 |- F/ y ph
2		nfv 1466	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1509	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1863	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2858	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1509	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1701	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4227	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2845	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 232	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2586	. 2 |- ( A. x e. om ( ph -> [. suc x / x ]. ph ) <-> A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) )
12		bj-bdfindes.bd	. . 3 |- BOUNDED ph
13		nfsbc1v 2858	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 2849	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 156	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1698	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 2849	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
18	17	biimprd 156	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
19	12, 13, 4, 5, 15, 16, 18	bj-bdfindis 11725	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
20	11, 19	sylan2b 281	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   [wsb 1692   A.wral 2359   [.wsbc 2840   (/)c0 3286   suc csuc 4190   omcom 4403  BOUNDED wbd 11586

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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963  ax-pr 4034  ax-un 4258  ax-bd0 11587  ax-bdor 11590  ax-bdex 11593  ax-bdeq 11594  ax-bdel 11595  ax-bdsb 11596  ax-bdsep 11658  ax-infvn 11719

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3450  df-pr 3451  df-uni 3652  df-int 3687  df-suc 4196  df-iom 4404  df-bdc 11615  df-bj-ind 11705

This theorem is referenced by: (None)