Theorem nfsbt 1950

Description: Closed form of nfsb 1920. (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 1507	. 2 |- ( A. x F/ z ph -> A. w A. x F/ z ph )
2		nfsbxyt 1917	. . . . 5 |- ( A. x F/ z ph -> F/ z [ w / x ] ph )
3	2	alimi 1432	. . . 4 |- ( A. w A. x F/ z ph -> A. w F/ z [ w / x ] ph )
4		nfsbxyt 1917	. . . 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 1509	. . . . 5 |- F/ w ph
7	6	sbco2 1939	. . . 4 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	7	nfbii 1450	. . 3 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
9	5, 8	sylib 121	. 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 1330   F/wnf 1437   [wsb 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  nfsbd  1951  setindft  13334