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Theorem setindft 13247
Description: Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
Assertion
Ref Expression
setindft  |-  ( A. x F/ y ph  ->  ( A. x ( A. y  e.  x  [
y  /  x ] ph  ->  ph )  ->  A. x ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem setindft
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfa1 1521 . . 3  |-  F/ x A. x F/ y ph
2 nfv 1508 . . . . . 6  |-  F/ z A. x F/ y
ph
3 nfnf1 1523 . . . . . . 7  |-  F/ y F/ y ph
43nfal 1555 . . . . . 6  |-  F/ y A. x F/ y
ph
5 nfsbt 1949 . . . . . 6  |-  ( A. x F/ y ph  ->  F/ y [ z  /  x ] ph )
6 nfv 1508 . . . . . . 7  |-  F/ z [ y  /  x ] ph
76a1i 9 . . . . . 6  |-  ( A. x F/ y ph  ->  F/ z [ y  /  x ] ph )
8 sbequ 1812 . . . . . . 7  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
98a1i 9 . . . . . 6  |-  ( A. x F/ y ph  ->  ( z  =  y  -> 
( [ z  /  x ] ph  <->  [ y  /  x ] ph )
) )
102, 4, 5, 7, 9cbvrald 13079 . . . . 5  |-  ( A. x F/ y ph  ->  ( A. z  e.  x  [ z  /  x ] ph  <->  A. y  e.  x  [ y  /  x ] ph ) )
1110biimpd 143 . . . 4  |-  ( A. x F/ y ph  ->  ( A. z  e.  x  [ z  /  x ] ph  ->  A. y  e.  x  [ y  /  x ] ph )
)
1211imim1d 75 . . 3  |-  ( A. x F/ y ph  ->  ( ( A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  ( A. z  e.  x  [ z  /  x ] ph  ->  ph )
) )
131, 12alimd 1501 . 2  |-  ( A. x F/ y ph  ->  ( A. x ( A. y  e.  x  [
y  /  x ] ph  ->  ph )  ->  A. x
( A. z  e.  x  [ z  /  x ] ph  ->  ph )
) )
14 ax-setind 4452 . 2  |-  ( A. x ( A. z  e.  x  [ z  /  x ] ph  ->  ph )  ->  A. x ph )
1513, 14syl6 33 1  |-  ( A. x F/ y ph  ->  ( A. x ( A. y  e.  x  [
y  /  x ] ph  ->  ph )  ->  A. x ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329   F/wnf 1436   [wsb 1735   A.wral 2416
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-ral 2421
This theorem is referenced by:  setindf  13248
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