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Theorem bj-bdfindes 13984
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13982 for explanations. From this version, it is easy to prove the bounded version of findes 4587. (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 1521 . . . 4  |-  F/ y
ph
2 nfv 1521 . . . 4  |-  F/ y
[. suc  x  /  x ]. ph
31, 2nfim 1565 . . 3  |-  F/ y ( ph  ->  [. suc  x  /  x ]. ph )
4 nfs1v 1932 . . . 4  |-  F/ x [ y  /  x ] ph
5 nfsbc1v 2973 . . . 4  |-  F/ x [. suc  y  /  x ]. ph
64, 5nfim 1565 . . 3  |-  F/ x
( [ y  /  x ] ph  ->  [. suc  y  /  x ]. ph )
7 sbequ12 1764 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 suceq 4387 . . . . 5  |-  ( x  =  y  ->  suc  x  =  suc  y )
98sbceq1d 2960 . . . 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 2692 . 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 2973 . . 3  |-  F/ x [. (/)  /  x ]. ph
14 sbceq1a 2964 . . . 4  |-  ( x  =  (/)  ->  ( ph  <->  [. (/)  /  x ]. ph )
)
1514biimprd 157 . . 3  |-  ( x  =  (/)  ->  ( [. (/)  /  x ]. ph  ->  ph ) )
16 sbequ1 1761 . . 3  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
17 sbceq1a 2964 . . . 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 13982 . 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 1348   [wsb 1755   A.wral 2448   [.wsbc 2955   (/)c0 3414   suc csuc 4350   omcom 4574  BOUNDED wbd 13847
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4115  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919  ax-infvn 13976
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-suc 4356  df-iom 4575  df-bdc 13876  df-bj-ind 13962
This theorem is referenced by: (None)
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