Theorem nfsbt 1891

Description: Closed form of nfsb 1863. (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 1459	. 2 |- ( A. x F/ z ph -> A. w A. x F/ z ph )
2		nfsbxyt 1860	. . . . 5 |- ( A. x F/ z ph -> F/ z [ w / x ] ph )
3	2	alimi 1384	. . . 4 |- ( A. w A. x F/ z ph -> A. w F/ z [ w / x ] ph )
4		nfsbxyt 1860	. . . 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 1461	. . . . 5 |- F/ w ph
7	6	sbco2 1880	. . . 4 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	nfbii 1402	. . 3 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
9	5, 8	sylib 120	. 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 1282   F/wnf 1389   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  nfsbd  1892  setindft  10760