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Theorem bj-bdfindes 13595
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13593 for explanations. From this version, it is easy to prove the bounded version of findes 4564. (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 1508 . . . 4  |-  F/ y
ph
2 nfv 1508 . . . 4  |-  F/ y
[. suc  x  /  x ]. ph
31, 2nfim 1552 . . 3  |-  F/ y ( ph  ->  [. suc  x  /  x ]. ph )
4 nfs1v 1919 . . . 4  |-  F/ x [ y  /  x ] ph
5 nfsbc1v 2955 . . . 4  |-  F/ x [. suc  y  /  x ]. ph
64, 5nfim 1552 . . 3  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
7 sbequ12 1751 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 suceq 4364 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
98sbceq1d 2942 . . . 4  |-  ( x  =  y  ->  ( [. suc  x  /  x ]. ph  <->  [. suc  y  /  x ]. ph ) )
107, 9imbi12d 233 . . 3  |-  ( x  =  y  ->  (
( ph  ->  [. suc  x  /  x ]. ph )  <->  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
) )
113, 6, 10cbvral 2676 . 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 2955 . . 3  |-  F/ x [. (/)  /  x ]. ph
14 sbceq1a 2946 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  [. (/)  /  x ]. ph )
)
1514biimprd 157 . . 3  |-  ( x  =  (/)  ->  ( [. (/)  /  x ]. ph  ->  ph ) )
16 sbequ1 1748 . . 3  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
17 sbceq1a 2946 . . . 4  |-  ( x  =  suc  y  -> 
( ph  <->  [. suc  y  /  x ]. ph ) )
1817biimprd 157 . . 3  |-  ( x  =  suc  y  -> 
( [. suc  y  /  x ]. ph  ->  ph )
)
1912, 13, 4, 5, 15, 16, 18bj-bdfindis 13593 . 2  |-  ( (
[. (/)  /  x ]. ph 
/\  A. y  e.  om  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)  ->  A. x  e.  om  ph )
2011, 19sylan2b 285 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 103    = wceq 1335   [wsb 1742   A.wral 2435   [.wsbc 2937   (/)c0 3395   suc csuc 4327   omcom 4551  BOUNDED wbd 13458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4092  ax-pr 4171  ax-un 4395  ax-bd0 13459  ax-bdor 13462  ax-bdex 13465  ax-bdeq 13466  ax-bdel 13467  ax-bdsb 13468  ax-bdsep 13530  ax-infvn 13587
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-sn 3567  df-pr 3568  df-uni 3775  df-int 3810  df-suc 4333  df-iom 4552  df-bdc 13487  df-bj-ind 13573
This theorem is referenced by: (None)
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