Theorem bj-bdfindes 16719

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 16717 for explanations. From this version, it is easy to prove the bounded version of findes 4725. (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 1577	. . . 4 |- F/ y ph
2		nfv 1577	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1621	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1993	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3061	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1621	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1820	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4523	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 3047	. . . 4 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
10	7, 9	imbi12d 234	. . 3 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
11	3, 6, 10	cbvral 2774	. 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 3061	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 3052	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1817	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 3052	. . . 4 |- ( x = suc y -> ( ph <-> [. suc y / x ]. ph ) )
18	17	biimprd 158	. . 3 |- ( x = suc y -> ( [. suc y / x ]. ph -> ph ) )
19	12, 13, 4, 5, 15, 16, 18	bj-bdfindis 16717	. 2 |- ( ( [. (/) / x ]. ph /\ A. y e. om ( [ y / x ] ph -> [. suc y / x ]. ph ) ) -> A. x e. om ph )
20	11, 19	sylan2b 287	1 |- ( ( [. (/) / x ]. ph /\ A. x e. om ( ph -> [. suc x / x ]. ph ) ) -> A. x e. om ph )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   [wsb 1811   A.wral 2520   [.wsbc 3042   (/)c0 3508   suc csuc 4486   omcom 4712  BOUNDED wbd 16582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-nul 4236  ax-pr 4322  ax-un 4554  ax-bd0 16583  ax-bdor 16586  ax-bdex 16589  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592  ax-bdsep 16654  ax-infvn 16711

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-suc 4492  df-iom 4713  df-bdc 16611  df-bj-ind 16697

This theorem is referenced by: (None)