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Theorem setindf 13333
Description: Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
Hypothesis
Ref Expression
setindf.nf  |-  F/ y
ph
Assertion
Ref Expression
setindf  |-  ( 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 setindf
StepHypRef Expression
1 setindft 13332 . 2  |-  ( A. x F/ y ph  ->  ( A. x ( A. y  e.  x  [
y  /  x ] ph  ->  ph )  ->  A. x ph ) )
2 setindf.nf . 2  |-  F/ y
ph
31, 2mpg 1428 1  |-  ( A. x ( A. y  e.  x  [ y  /  x ] ph  ->  ph )  ->  A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330   F/wnf 1437   [wsb 1736   A.wral 2417
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-setind 4459
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-ral 2422
This theorem is referenced by: (None)
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