Theorem nfsbt 2005

Description: Closed form of nfsb 1975. (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 1550	. 2 |- ( A. x F/ z ph -> A. w A. x F/ z ph )
2		nfsbxyt 1972	. . . . 5 |- ( A. x F/ z ph -> F/ z [ w / x ] ph )
3	2	alimi 1479	. . . 4 |- ( A. w A. x F/ z ph -> A. w F/ z [ w / x ] ph )
4		nfsbxyt 1972	. . . 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 1552	. . . . 5 |- F/ w ph
7	6	sbco2 1994	. . . 4 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	nfbii 1497	. . 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 1371   F/wnf 1484   [wsb 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  nfsbd  2006  setindft  16100