Theorem bj-bdfindis 13316

Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4522 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4522, finds2 4523, finds1 4524. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-bdfindis.bd	|- BOUNDED ph
bj-bdfindis.nf0	|- F/ x ps
bj-bdfindis.nf1	|- F/ x ch
bj-bdfindis.nfsuc	|- F/ x th
bj-bdfindis.0	|- ( x = (/) -> ( ps -> ph ) )
bj-bdfindis.1	|- ( x = y -> ( ph -> ch ) )
bj-bdfindis.suc	|- ( x = suc y -> ( th -> ph ) )

Assertion

Ref	Expression
bj-bdfindis	|- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )

Distinct variable groups:   x, y   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)   ch( x, y)   th( x, y)

Proof of Theorem bj-bdfindis

Step	Hyp	Ref	Expression
1		bj-bdfindis.nf0	. . . 4 |- F/ x ps
2		0ex 4063	. . . 4 |- (/) e. _V
3		bj-bdfindis.0	. . . 4 |- ( x = (/) -> ( ps -> ph ) )
4	1, 2, 3	elabf2 13160	. . 3 |- ( ps -> (/) e. { x | ph } )
5		bj-bdfindis.nf1	. . . . . 6 |- F/ x ch
6		bj-bdfindis.1	. . . . . 6 |- ( x = y -> ( ph -> ch ) )
7	5, 6	elabf1 13159	. . . . 5 |- ( y e. { x | ph } -> ch )
8		bj-bdfindis.nfsuc	. . . . . 6 |- F/ x th
9		vex 2692	. . . . . . 7 |- y e. _V
10	9	bj-sucex 13292	. . . . . 6 |- suc y e. _V
11		bj-bdfindis.suc	. . . . . 6 |- ( x = suc y -> ( th -> ph ) )
12	8, 10, 11	elabf2 13160	. . . . 5 |- ( th -> suc y e. { x | ph } )
13	7, 12	imim12i 59	. . . 4 |- ( ( ch -> th ) -> ( y e. { x | ph } -> suc y e. { x | ph } ) )
14	13	ralimi 2498	. . 3 |- ( A. y e. om ( ch -> th ) -> A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } ) )
15		bj-bdfindis.bd	. . . . 5 |- BOUNDED ph
16	15	bdcab 13218	. . . 4 |- BOUNDED { x | ph }
17	16	bdpeano5 13312	. . 3 |- ( ( (/) e. { x | ph } /\ A. y e. om ( y e. { x | ph } -> suc y e. { x | ph } ) ) -> om C_ { x | ph } )
18	4, 14, 17	syl2an 287	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> om C_ { x | ph } )
19		ssabral 3173	. 2 |- ( om C_ { x | ph } <-> A. x e. om ph )
20	18, 19	sylib 121	1 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   F/wnf 1437   e. wcel 1481   {cab 2126   A.wral 2417   C_ wss 3076   (/)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-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:  bj-bdfindisg  13317  bj-bdfindes  13318  bj-nn0suc0  13319