Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  setindf Unicode version

Theorem setindf 14945
Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness 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 14944 . 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 1461 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 1361   F/wnf 1470   [wsb 1772   A.wral 2465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-ral 2470
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator