Theorem bj-bdfindis 10885

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 4343 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4343, finds2 4344, finds1 4345. (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 3907	. . . 4 |- (/) e. _V
3		bj-bdfindis.0	. . . 4 |- ( x = (/) -> ( ps -> ph ) )
4	1, 2, 3	elabf2 10728	. . 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 10727	. . . . 5 |- ( y e. { x | ph } -> ch )
8		bj-bdfindis.nfsuc	. . . . . 6 |- F/ x th
9		vex 2605	. . . . . . 7 |- y e. _V
10	9	bj-sucex 10857	. . . . . 6 |- suc y e. _V
11		bj-bdfindis.suc	. . . . . 6 |- ( x = suc y -> ( th -> ph ) )
12	8, 10, 11	elabf2 10728	. . . . 5 |- ( th -> suc y e. { x | ph } )
13	7, 12	imim12i 58	. . . 4 |- ( ( ch -> th ) -> ( y e. { x | ph } -> suc y e. { x | ph } ) )
14	13	ralimi 2427	. . 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 10783	. . . 4 |- BOUNDED { x | ph }
17	16	bdpeano5 10881	. . 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 283	. 2 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> om C_ { x | ph } )
19		ssabral 3066	. 2 |- ( om C_ { x | ph } <-> A. x e. om ph )
20	18, 19	sylib 120	1 |- ( ( ps /\ A. y e. om ( ch -> th ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   F/wnf 1390   e. wcel 1434   {cab 2068   A.wral 2349   C_ wss 2974   (/)c0 3252   suc csuc 4122   omcom 4333  BOUNDED wbd 10746

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-nul 3906  ax-pr 3966  ax-un 4190  ax-bd0 10747  ax-bdor 10750  ax-bdex 10753  ax-bdeq 10754  ax-bdel 10755  ax-bdsb 10756  ax-bdsep 10818  ax-infvn 10879

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-pr 3407  df-uni 3604  df-int 3639  df-suc 4128  df-iom 4334  df-bdc 10775  df-bj-ind 10865

This theorem is referenced by:  bj-bdfindisg  10886  bj-bdfindes  10887  bj-nn0suc0  10888