Theorem nfsbt 1995

Description: Closed form of nfsb 1965. (Contributed by Jim Kingdon, 9-May-2018.)

Assertion

Ref	Expression
nfsbt	|- ( A. x F/ z ph -> F/ z [ y / x ] ph )

Distinct variable group:   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem nfsbt

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1540	. 2 |- ( A. x F/ z ph -> A. w A. x F/ z ph )
2		nfsbxyt 1962	. . . . 5 |- ( A. x F/ z ph -> F/ z [ w / x ] ph )
3	2	alimi 1469	. . . 4 |- ( A. w A. x F/ z ph -> A. w F/ z [ w / x ] ph )
4		nfsbxyt 1962	. . . 4 |- ( A. w F/ z [ w / x ] ph -> F/ z [ y / w ] [ w / x ] ph )
5	3, 4	syl 14	. . 3 |- ( A. w A. x F/ z ph -> F/ z [ y / w ] [ w / x ] ph )
6		nfv 1542	. . . . 5 |- F/ w ph
7	6	sbco2 1984	. . . 4 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	nfbii 1487	. . 3 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
9	5, 8	sylib 122	. 2 |- ( A. w A. x F/ z ph -> F/ z [ y / x ] ph )
10	1, 9	syl 14	1 |- ( A. x F/ z ph -> F/ z [ y / x ] ph )

Syntax hints:   -> wi 4   A.wal 1362   F/wnf 1474   [wsb 1776

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

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  nfsbd  1996  setindft  15611