Theorem setindft 12164

Description: Axiom of set-induction with a disjoint variable condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.)

Assertion

Ref	Expression
setindft	|- ( A. x F/ y ph -> ( 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 setindft

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1480	. . 3 |- F/ x A. x F/ y ph
2		nfv 1467	. . . . . 6 |- F/ z A. x F/ y ph
3		nfnf1 1482	. . . . . . 7 |- F/ y F/ y ph
4	3	nfal 1514	. . . . . 6 |- F/ y A. x F/ y ph
5		nfsbt 1899	. . . . . 6 |- ( A. x F/ y ph -> F/ y [ z / x ] ph )
6		nfv 1467	. . . . . . 7 |- F/ z [ y / x ] ph
7	6	a1i 9	. . . . . 6 |- ( A. x F/ y ph -> F/ z [ y / x ] ph )
8		sbequ 1769	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
9	8	a1i 9	. . . . . 6 |- ( A. x F/ y ph -> ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) ) )
10	2, 4, 5, 7, 9	cbvrald 11992	. . . . 5 |- ( A. x F/ y ph -> ( A. z e. x [ z / x ] ph <-> A. y e. x [ y / x ] ph ) )
11	10	biimpd 143	. . . 4 |- ( A. x F/ y ph -> ( A. z e. x [ z / x ] ph -> A. y e. x [ y / x ] ph ) )
12	11	imim1d 75	. . 3 |- ( A. x F/ y ph -> ( ( A. y e. x [ y / x ] ph -> ph ) -> ( A. z e. x [ z / x ] ph -> ph ) ) )
13	1, 12	alimd 1460	. 2 |- ( A. x F/ y ph -> ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ( A. z e. x [ z / x ] ph -> ph ) ) )
14		ax-setind 4368	. 2 |- ( A. x ( A. z e. x [ z / x ] ph -> ph ) -> A. x ph )
15	13, 14	syl6 33	1 |- ( A. x F/ y ph -> ( A. x ( A. y e. x [ y / x ] ph -> ph ) -> A. x ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1288   F/wnf 1395   [wsb 1693   A.wral 2360

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4368

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-ral 2365

This theorem is referenced by:  setindf  12165