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Theorem bj-bdfindes 12981
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 12979 for explanations. From this version, it is easy to prove the bounded version of findes 4485. (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 1491 . . . 4  |-  F/ y
ph
2 nfv 1491 . . . 4  |-  F/ y
[. suc  x  /  x ]. ph
31, 2nfim 1534 . . 3  |-  F/ y ( ph  ->  [. suc  x  /  x ]. ph )
4 nfs1v 1890 . . . 4  |-  F/ x [ y  /  x ] ph
5 nfsbc1v 2898 . . . 4  |-  F/ x [. suc  y  /  x ]. ph
64, 5nfim 1534 . . 3  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
7 sbequ12 1727 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 suceq 4292 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
98sbceq1d 2885 . . . 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 2625 . 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 2898 . . 3  |-  F/ x [. (/)  /  x ]. ph
14 sbceq1a 2889 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  [. (/)  /  x ]. ph )
)
1514biimprd 157 . . 3  |-  ( x  =  (/)  ->  ( [. (/)  /  x ]. ph  ->  ph ) )
16 sbequ1 1724 . . 3  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
17 sbceq1a 2889 . . . 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 12979 . 2  |-  ( (
[. (/)  /  x ]. ph 
/\  A. y  e.  om  ( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
)  ->  A. x  e.  om  ph )
2011, 19sylan2b 283 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 1314   [wsb 1718   A.wral 2391   [.wsbc 2880   (/)c0 3331   suc csuc 4255   omcom 4472  BOUNDED wbd 12844
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12845  ax-bdor 12848  ax-bdex 12851  ax-bdeq 12852  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916  ax-infvn 12973
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12873  df-bj-ind 12959
This theorem is referenced by: (None)
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