Theorem bj-bdfindes 13318

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13316 for explanations. From this version, it is easy to prove the bounded version of findes 4525. (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 1509	. . . 4 |- F/ y ph
2		nfv 1509	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1552	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1913	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2931	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1552	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1745	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4332	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2918	. . . 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 2653	. 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 2931	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 2922	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 157	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1742	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 2922	. . . 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 13316	. 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 1332   [wsb 1736   A.wral 2417   [.wsbc 2913   (/)c0 3368   suc csuc 4295   omcom 4512  BOUNDED wbd 13181

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062  ax-pr 4139  ax-un 4363  ax-bd0 13182  ax-bdor 13185  ax-bdex 13188  ax-bdeq 13189  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253  ax-infvn 13310

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-iom 4513  df-bdc 13210  df-bj-ind 13296

This theorem is referenced by: (None)