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Theorem setindf 9350
Description: Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)
Hypothesis
Ref Expression
setindf.nf yφ
Assertion
Ref Expression
setindf (x(y x [y / x]φφ) → xφ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem setindf
StepHypRef Expression
1 setindft 9349 . 2 (xyφ → (x(y x [y / x]φφ) → xφ))
2 setindf.nf . 2 yφ
31, 2mpg 1337 1 (x(y x [y / x]φφ) → xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wnf 1346  [wsb 1642  wral 2300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-cleq 2030  df-clel 2033  df-ral 2305
This theorem is referenced by: (None)
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