Theorem setindf 16040

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 16039	. 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 1475	1 |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1371   F/wnf 1484   [wsb 1786   A.wral 2485

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202  df-ral 2490

This theorem is referenced by: (None)