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Theorem bj-bdfindes 13074
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13072 for explanations. From this version, it is easy to prove the bounded version of findes 4487. (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 1493 . . . 4  |-  F/ y
ph
2 nfv 1493 . . . 4  |-  F/ y
[. suc  x  /  x ]. ph
31, 2nfim 1536 . . 3  |-  F/ y ( ph  ->  [. suc  x  /  x ]. ph )
4 nfs1v 1892 . . . 4  |-  F/ x [ y  /  x ] ph
5 nfsbc1v 2900 . . . 4  |-  F/ x [. suc  y  /  x ]. ph
64, 5nfim 1536 . . 3  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
7 sbequ12 1729 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 suceq 4294 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
98sbceq1d 2887 . . . 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 2627 . 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 2900 . . 3  |-  F/ x [. (/)  /  x ]. ph
14 sbceq1a 2891 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  [. (/)  /  x ]. ph )
)
1514biimprd 157 . . 3  |-  ( x  =  (/)  ->  ( [. (/)  /  x ]. ph  ->  ph ) )
16 sbequ1 1726 . . 3  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
17 sbceq1a 2891 . . . 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 13072 . 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 1316   [wsb 1720   A.wral 2393   [.wsbc 2882   (/)c0 3333   suc csuc 4257   omcom 4474  BOUNDED wbd 12937
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-nul 4024  ax-pr 4101  ax-un 4325  ax-bd0 12938  ax-bdor 12941  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947  ax-bdsep 13009  ax-infvn 13066
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-pr 3504  df-uni 3707  df-int 3742  df-suc 4263  df-iom 4475  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by: (None)
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