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Theorem setindis 12750
Description: Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.)
Hypotheses
Ref Expression
setindis.nf0  |-  F/ x ps
setindis.nf1  |-  F/ x ch
setindis.nf2  |-  F/ y
ph
setindis.nf3  |-  F/ y ps
setindis.1  |-  ( x  =  z  ->  ( ph  ->  ps ) )
setindis.2  |-  ( x  =  y  ->  ( ch  ->  ph ) )
Assertion
Ref Expression
setindis  |-  ( 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 setindis
StepHypRef Expression
1 nfcv 2240 . . . . 5  |-  F/_ x
y
2 setindis.nf0 . . . . 5  |-  F/ x ps
31, 2nfralxy 2430 . . . 4  |-  F/ x A. z  e.  y  ps
4 setindis.nf1 . . . 4  |-  F/ x ch
53, 4nfim 1519 . . 3  |-  F/ x
( A. z  e.  y  ps  ->  ch )
6 nfcv 2240 . . . . 5  |-  F/_ y
x
7 setindis.nf3 . . . . 5  |-  F/ y ps
86, 7nfralxy 2430 . . . 4  |-  F/ y A. z  e.  x  ps
9 setindis.nf2 . . . 4  |-  F/ y
ph
108, 9nfim 1519 . . 3  |-  F/ y ( A. z  e.  x  ps  ->  ph )
11 raleq 2584 . . . . 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 setindis.2 . . . . 5  |-  ( x  =  y  ->  ( ch  ->  ph ) )
1413equcoms 1652 . . . 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 1688 . 2  |-  ( A. y ( A. z  e.  y  ps  ->  ch )  ->  A. x
( A. z  e.  x  ps  ->  ph )
)
17 setindis.1 . . . . . 6  |-  ( x  =  z  ->  ( ph  ->  ps ) )
182, 17bj-sbime 12562 . . . . 5  |-  ( [ z  /  x ] ph  ->  ps )
1918ralimi 2454 . . . 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 1399 . 2  |-  ( A. x ( A. z  e.  x  ps  ->  ph )  ->  A. x
( A. z  e.  x  [ z  /  x ] ph  ->  ph )
)
22 ax-setind 4390 . 2  |-  ( A. x ( A. z  e.  x  [ z  /  x ] ph  ->  ph )  ->  A. x ph )
2316, 21, 223syl 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 1297   F/wnf 1404   [wsb 1703   A.wral 2375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380
This theorem is referenced by:  bj-inf2vnlem4  12756  bj-findis  12762
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