Theorem nfsbt 1986

Description: Closed form of nfsb 1956. (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 1536	. 2 |- ( A. x F/ z ph -> A. w A. x F/ z ph )
2		nfsbxyt 1953	. . . . 5 |- ( A. x F/ z ph -> F/ z [ w / x ] ph )
3	2	alimi 1465	. . . 4 |- ( A. w A. x F/ z ph -> A. w F/ z [ w / x ] ph )
4		nfsbxyt 1953	. . . 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 1538	. . . . 5 |- F/ w ph
7	6	sbco2 1975	. . . 4 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	nfbii 1483	. . 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 1361   F/wnf 1470   [wsb 1772

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773

This theorem is referenced by:  nfsbd  1987  setindft  15013