Theorem setindf 16561

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 16560	. 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 1499	1 |- ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1395   F/wnf 1508   [wsb 1810   A.wral 2510

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-cleq 2224  df-clel 2227  df-ral 2515

This theorem is referenced by: (None)