Theorem setindft 14802

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 1541	. . 3 |- F/ x A. x F/ y ph
2		nfv 1528	. . . . . 6 |- F/ z A. x F/ y ph
3		nfnf1 1544	. . . . . . 7 |- F/ y F/ y ph
4	3	nfal 1576	. . . . . 6 |- F/ y A. x F/ y ph
5		nfsbt 1976	. . . . . 6 |- ( A. x F/ y ph -> F/ y [ z / x ] ph )
6		nfv 1528	. . . . . . 7 |- F/ z [ y / x ] ph
7	6	a1i 9	. . . . . 6 |- ( A. x F/ y ph -> F/ z [ y / x ] ph )
8		sbequ 1840	. . . . . . 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 14625	. . . . 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 1521	. 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 4538	. 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 1351   F/wnf 1460   [wsb 1762   A.wral 2455

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-ral 2460

This theorem is referenced by:  setindf  14803