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Theorem setindft 10477
Description: Axiom of set-induction with a DV 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 1450 . . 3  |-  F/ x A. x F/ y ph
2 nfv 1437 . . . . . 6  |-  F/ z A. x F/ y
ph
3 nfnf1 1452 . . . . . . 7  |-  F/ y F/ y ph
43nfal 1484 . . . . . 6  |-  F/ y A. x F/ y
ph
5 nfsbt 1866 . . . . . 6  |-  ( A. x F/ y ph  ->  F/ y [ z  /  x ] ph )
6 nfv 1437 . . . . . . 7  |-  F/ z [ y  /  x ] ph
76a1i 9 . . . . . 6  |-  ( A. x F/ y ph  ->  F/ z [ y  /  x ] ph )
8 sbequ 1737 . . . . . . 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 10314 . . . . 5  |-  ( A. x F/ y ph  ->  ( A. z  e.  x  [ z  /  x ] ph  <->  A. y  e.  x  [ y  /  x ] ph ) )
1110biimpd 136 . . . 4  |-  ( A. x F/ y ph  ->  ( A. z  e.  x  [ z  /  x ] ph  ->  A. y  e.  x  [ y  /  x ] ph )
)
1211imim1d 73 . . 3  |-  ( A. x F/ y ph  ->  ( ( A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  ( A. z  e.  x  [ z  /  x ] ph  ->  ph )
) )
131, 12alimd 1430 . 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 4290 . 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 102   A.wal 1257   F/wnf 1365   [wsb 1661   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052  df-ral 2328
This theorem is referenced by:  setindf  10478
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