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Theorem bj-bdfindis 14355
Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4596 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4596, finds2 4597, finds1 4598. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd  |- BOUNDED  ph
bj-bdfindis.nf0  |-  F/ x ps
bj-bdfindis.nf1  |-  F/ x ch
bj-bdfindis.nfsuc  |-  F/ x th
bj-bdfindis.0  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
bj-bdfindis.1  |-  ( x  =  y  ->  ( ph  ->  ch ) )
bj-bdfindis.suc  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
Assertion
Ref Expression
bj-bdfindis  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
Distinct variable groups:    x, y    ph, y
Allowed substitution hints:    ph( x)    ps( x, y)    ch( x, y)    th( x, y)

Proof of Theorem bj-bdfindis
StepHypRef Expression
1 bj-bdfindis.nf0 . . . 4  |-  F/ x ps
2 0ex 4127 . . . 4  |-  (/)  e.  _V
3 bj-bdfindis.0 . . . 4  |-  ( x  =  (/)  ->  ( ps 
->  ph ) )
41, 2, 3elabf2 14190 . . 3  |-  ( ps 
->  (/)  e.  { x  |  ph } )
5 bj-bdfindis.nf1 . . . . . 6  |-  F/ x ch
6 bj-bdfindis.1 . . . . . 6  |-  ( x  =  y  ->  ( ph  ->  ch ) )
75, 6elabf1 14189 . . . . 5  |-  ( y  e.  { x  | 
ph }  ->  ch )
8 bj-bdfindis.nfsuc . . . . . 6  |-  F/ x th
9 vex 2740 . . . . . . 7  |-  y  e. 
_V
109bj-sucex 14331 . . . . . 6  |-  suc  y  e.  _V
11 bj-bdfindis.suc . . . . . 6  |-  ( x  =  suc  y  -> 
( th  ->  ph )
)
128, 10, 11elabf2 14190 . . . . 5  |-  ( th 
->  suc  y  e.  {
x  |  ph }
)
137, 12imim12i 59 . . . 4  |-  ( ( ch  ->  th )  ->  ( y  e.  {
x  |  ph }  ->  suc  y  e.  {
x  |  ph }
) )
1413ralimi 2540 . . 3  |-  ( A. y  e.  om  ( ch  ->  th )  ->  A. y  e.  om  ( y  e. 
{ x  |  ph }  ->  suc  y  e.  { x  |  ph }
) )
15 bj-bdfindis.bd . . . . 5  |- BOUNDED  ph
1615bdcab 14257 . . . 4  |- BOUNDED  { x  |  ph }
1716bdpeano5 14351 . . 3  |-  ( (
(/)  e.  { x  |  ph }  /\  A. y  e.  om  (
y  e.  { x  |  ph }  ->  suc  y  e.  { x  |  ph } ) )  ->  om  C_  { x  |  ph } )
184, 14, 17syl2an 289 . 2  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  om  C_  { x  |  ph } )
19 ssabral 3226 . 2  |-  ( om  C_  { x  |  ph } 
<-> 
A. x  e.  om  ph )
2018, 19sylib 122 1  |-  ( ( ps  /\  A. y  e.  om  ( ch  ->  th ) )  ->  A. x  e.  om  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   F/wnf 1460    e. wcel 2148   {cab 2163   A.wral 2455    C_ wss 3129   (/)c0 3422   suc csuc 4362   omcom 4586  BOUNDED wbd 14220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4126  ax-pr 4206  ax-un 4430  ax-bd0 14221  ax-bdor 14224  ax-bdex 14227  ax-bdeq 14228  ax-bdel 14229  ax-bdsb 14230  ax-bdsep 14292  ax-infvn 14349
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-suc 4368  df-iom 4587  df-bdc 14249  df-bj-ind 14335
This theorem is referenced by:  bj-bdfindisg  14356  bj-bdfindes  14357  bj-nn0suc0  14358
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