Theorem bj-bdfindes 15595

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 15593 for explanations. From this version, it is easy to prove the bounded version of findes 4639. (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 1542	. . . 4 |- F/ y ph
2		nfv 1542	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1586	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1958	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 3008	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1586	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1785	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4437	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2994	. . . 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 2725	. 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 3008	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 2999	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1782	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 2999	. . . 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 15593	. 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 1364   [wsb 1776   A.wral 2475   [.wsbc 2989   (/)c0 3450   suc csuc 4400   omcom 4626  BOUNDED wbd 15458

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4159  ax-pr 4242  ax-un 4468  ax-bd0 15459  ax-bdor 15462  ax-bdex 15465  ax-bdeq 15466  ax-bdel 15467  ax-bdsb 15468  ax-bdsep 15530  ax-infvn 15587

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627  df-bdc 15487  df-bj-ind 15573

This theorem is referenced by: (None)