Theorem bj-bdfindes 13595

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13593 for explanations. From this version, it is easy to prove the bounded version of findes 4564. (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 1508	. . . 4 |- F/ y ph
2		nfv 1508	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1552	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1919	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2955	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1552	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1751	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4364	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2942	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 233	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2676	. 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 2955	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 2946	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 157	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1748	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 2946	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
18	17	biimprd 157	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
19	12, 13, 4, 5, 15, 16, 18	bj-bdfindis 13593	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
20	11, 19	sylan2b 285	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   [wsb 1742   A.wral 2435   [.wsbc 2937   (/)c0 3395   suc csuc 4327   omcom 4551  BOUNDED wbd 13458

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4092  ax-pr 4171  ax-un 4395  ax-bd0 13459  ax-bdor 13462  ax-bdex 13465  ax-bdeq 13466  ax-bdel 13467  ax-bdsb 13468  ax-bdsep 13530  ax-infvn 13587

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568  df-uni 3775  df-int 3810  df-suc 4333  df-iom 4552  df-bdc 13487  df-bj-ind 13573

This theorem is referenced by: (None)