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Theorem bdsetindis 13336
Description: Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdsetindis.bd  |- BOUNDED  ph
bdsetindis.nf0  |-  F/ x ps
bdsetindis.nf1  |-  F/ x ch
bdsetindis.nf2  |-  F/ y
ph
bdsetindis.nf3  |-  F/ y ps
bdsetindis.1  |-  ( x  =  z  ->  ( ph  ->  ps ) )
bdsetindis.2  |-  ( x  =  y  ->  ( ch  ->  ph ) )
Assertion
Ref Expression
bdsetindis  |-  ( A. y ( A. z  e.  y  ps  ->  ch )  ->  A. x ph )
Distinct variable groups:    x, y, z    ph, z
Allowed substitution hints:    ph( x, y)    ps( x, y, z)    ch( x, y, z)

Proof of Theorem bdsetindis
StepHypRef Expression
1 nfcv 2282 . . . . 5  |-  F/_ x
y
2 bdsetindis.nf0 . . . . 5  |-  F/ x ps
31, 2nfralxy 2474 . . . 4  |-  F/ x A. z  e.  y  ps
4 bdsetindis.nf1 . . . 4  |-  F/ x ch
53, 4nfim 1552 . . 3  |-  F/ x
( A. z  e.  y  ps  ->  ch )
6 nfcv 2282 . . . . 5  |-  F/_ y
x
7 bdsetindis.nf3 . . . . 5  |-  F/ y ps
86, 7nfralxy 2474 . . . 4  |-  F/ y A. z  e.  x  ps
9 bdsetindis.nf2 . . . 4  |-  F/ y
ph
108, 9nfim 1552 . . 3  |-  F/ y ( A. z  e.  x  ps  ->  ph )
11 raleq 2629 . . . . 5  |-  ( y  =  x  ->  ( A. z  e.  y  ps 
<-> 
A. z  e.  x  ps ) )
1211biimprd 157 . . . 4  |-  ( y  =  x  ->  ( A. z  e.  x  ps  ->  A. z  e.  y  ps ) )
13 bdsetindis.2 . . . . 5  |-  ( x  =  y  ->  ( ch  ->  ph ) )
1413equcoms 1685 . . . 4  |-  ( y  =  x  ->  ( ch  ->  ph ) )
1512, 14imim12d 74 . . 3  |-  ( y  =  x  ->  (
( A. z  e.  y  ps  ->  ch )  ->  ( A. z  e.  x  ps  ->  ph ) ) )
165, 10, 15cbv3 1721 . 2  |-  ( A. y ( A. z  e.  y  ps  ->  ch )  ->  A. x
( A. z  e.  x  ps  ->  ph )
)
17 bdsetindis.1 . . . . . 6  |-  ( x  =  z  ->  ( ph  ->  ps ) )
182, 17bj-sbime 13149 . . . . 5  |-  ( [ z  /  x ] ph  ->  ps )
1918ralimi 2498 . . . 4  |-  ( A. z  e.  x  [
z  /  x ] ph  ->  A. z  e.  x  ps )
2019imim1i 60 . . 3  |-  ( ( A. z  e.  x  ps  ->  ph )  ->  ( A. z  e.  x  [ z  /  x ] ph  ->  ph ) )
2120alimi 1432 . 2  |-  ( A. x ( A. z  e.  x  ps  ->  ph )  ->  A. x
( A. z  e.  x  [ z  /  x ] ph  ->  ph )
)
22 bdsetindis.bd . . 3  |- BOUNDED  ph
2322ax-bdsetind 13335 . 2  |-  ( A. x ( A. z  e.  x  [ z  /  x ] ph  ->  ph )  ->  A. x ph )
2416, 21, 233syl 17 1  |-  ( A. y ( A. z  e.  y  ps  ->  ch )  ->  A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330   F/wnf 1437   [wsb 1736   A.wral 2417  BOUNDED wbd 13179
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bdsetind 13335
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422
This theorem is referenced by:  bj-inf2vnlem3  13339
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