Theorem setindf 12847

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

Step	Hyp	Ref	Expression
1		setindft 12846	. 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 1408	1 |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1310   F/wnf 1417   [wsb 1716   A.wral 2388

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-setind 4410

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-cleq 2106  df-clel 2109  df-ral 2393

This theorem is referenced by: (None)