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Theorem bj-bdfindes 11727
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 11725 for explanations. From this version, it is easy to prove the bounded version of findes 4416. (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.
StepHypRef Expression
1 nfv 1466 . . . 4  |-  F/ y
ph
2 nfv 1466 . . . 4  |-  F/ y
[. suc  x  /  x ]. ph
31, 2nfim 1509 . . 3  |-  F/ y ( ph  ->  [. suc  x  /  x ]. ph )
4 nfs1v 1863 . . . 4  |-  F/ x [ y  /  x ] ph
5 nfsbc1v 2858 . . . 4  |-  F/ x [. suc  y  /  x ]. ph
64, 5nfim 1509 . . 3  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
7 sbequ12 1701 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 suceq 4227 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
98sbceq1d 2845 . . . 4  |-  ( x  =  y  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  y  /  x ]. ph ) )
107, 9imbi12d 232 . . 3  |-  ( x  =  y  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
) )
113, 6, 10cbvral 2586 . 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 2858 . . 3  |-  F/ x [. (/)  /  x ]. ph
14 sbceq1a 2849 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  [. (/)  /  x ]. ph )
)
1514biimprd 156 . . 3  |-  ( x  =  (/)  ->  ( [. (/)  /  x ]. ph  ->  ph ) )
16 sbequ1 1698 . . 3  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
17 sbceq1a 2849 . . . 4  |-  ( x  =  suc  y  -> 
( ph  <->  [. suc  y  /  x ]. ph ) )
1817biimprd 156 . . 3  |-  ( x  =  suc  y  -> 
( [. suc  y  /  x ]. ph  ->  ph )
)
1912, 13, 4, 5, 15, 16, 18bj-bdfindis 11725 . 2  |-  ( (
[. (/)  /  x ]. ph 
/\  A. y  e.  om  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)  ->  A. x  e.  om  ph )
2011, 19sylan2b 281 1  |-  ( (
[. (/)  /  x ]. ph 
/\  A. x  e.  om  ( ph  ->  [. suc  x  /  x ]. ph )
)  ->  A. x  e.  om  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   [wsb 1692   A.wral 2359   [.wsbc 2840   (/)c0 3286   suc csuc 4190   omcom 4403  BOUNDED wbd 11586
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3963  ax-pr 4034  ax-un 4258  ax-bd0 11587  ax-bdor 11590  ax-bdex 11593  ax-bdeq 11594  ax-bdel 11595  ax-bdsb 11596  ax-bdsep 11658  ax-infvn 11719
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-sn 3450  df-pr 3451  df-uni 3652  df-int 3687  df-suc 4196  df-iom 4404  df-bdc 11615  df-bj-ind 11705
This theorem is referenced by: (None)
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