Theorem bj-bdfindes 14561

Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 14559 for explanations. From this version, it is easy to prove the bounded version of findes 4601. (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 1528	. . . 4 |- F/ y ph
2		nfv 1528	. . . 4 |- F/ y [. suc x / x ]. ph
3	1, 2	nfim 1572	. . 3 |- F/ y ( ph -> [. suc x / x ]. ph )
4		nfs1v 1939	. . . 4 |- F/ x [ y / x ] ph
5		nfsbc1v 2981	. . . 4 |- F/ x [. suc y / x ]. ph
6	4, 5	nfim 1572	. . 3 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
7		sbequ12 1771	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		suceq 4401	. . . . 5 |- ( x = y -> suc x = suc y )
9	8	sbceq1d 2967	. . . 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 2699	. 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 2981	. . 3 |- F/ x [. (/) / x ]. ph
14		sbceq1a 2972	. . . 4 |- ( x = (/) -> ( ph <-> [. (/) / x ]. ph ) )
15	14	biimprd 158	. . 3 |- ( x = (/) -> ( [. (/) / x ]. ph -> ph ) )
16		sbequ1 1768	. . 3 |- ( x = y -> ( ph -> [ y / x ] ph ) )
17		sbceq1a 2972	. . . 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 14559	. 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 1353   [wsb 1762   A.wral 2455   [.wsbc 2962   (/)c0 3422   suc csuc 4364   omcom 4588  BOUNDED wbd 14424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4128  ax-pr 4208  ax-un 4432  ax-bd0 14425  ax-bdor 14428  ax-bdex 14431  ax-bdeq 14432  ax-bdel 14433  ax-bdsb 14434  ax-bdsep 14496  ax-infvn 14553

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-suc 4370  df-iom 4589  df-bdc 14453  df-bj-ind 14539

This theorem is referenced by: (None)