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Theorem setindf 7184
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 7183 . 2 (xyφ → (x(y x [y / x]φφ) → xφ))
2 setindf.nf . 2 yφ
31, 2mpg 1320 1 (x(y x [y / x]φφ) → xφ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226  wnf 1329  [wsb 1627  wral 2284
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-cleq 2015  df-clel 2018  df-ral 2289
This theorem is referenced by: (None)
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