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

Step	Hyp	Ref	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
3	1, 2	mpg 1461	1 |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )

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)