Theorem setindft 15611

Description: Axiom of set-induction with a disjoint variable condition replaced with a nonfreeness 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 1555	. . 3 |- F/ x A. x F/ y ph
2		nfv 1542	. . . . . 6 |- F/ z A. x F/ y ph
3		nfnf1 1558	. . . . . . 7 |- F/ y F/ y ph
4	3	nfal 1590	. . . . . 6 |- F/ y A. x F/ y ph
5		nfsbt 1995	. . . . . 6 |- ( A. x F/ y ph -> F/ y [ z / x ] ph )
6		nfv 1542	. . . . . . 7 |- F/ z [ y / x ] ph
7	6	a1i 9	. . . . . 6 |- ( A. x F/ y ph -> F/ z [ y / x ] ph )
8		sbequ 1854	. . . . . . 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 15434	. . . . 5 |- ( A. x F/ y ph -> ( A. z e. x [ z / x ] ph <-> A. y e. x [ y / x ] ph ) )
11	10	biimpd 144	. . . 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 1535	. 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 4573	. 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 105   A.wal 1362   F/wnf 1474   [wsb 1776   A.wral 2475

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-ral 2480

This theorem is referenced by:  setindf  15612