Theorem nfsb 1919

Description: If z is not free in ph, it is not free in [ y / x ] ph when y and z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof rewritten by Jim Kingdon, 19-Mar-2018.)

Hypothesis

Ref	Expression
nfsb.1	|- F/ z ph

Assertion

Ref	Expression
nfsb	|- F/ z [ y / x ] ph

Distinct variable group:   y, z

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

Proof of Theorem nfsb

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfsb.1	. . . 4 |- F/ z ph
2	1	nfsbxy 1915	. . 3 |- F/ z [ w / x ] ph
3	2	nfsbxy 1915	. 2 |- F/ z [ y / w ] [ w / x ] ph
4		ax-17 1506	. . . 4 |- ( ph -> A. w ph )
5	4	sbco2vh 1918	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
6	5	nfbii 1449	. 2 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
7	3, 6	mpbi 144	1 |- F/ z [ y / x ] ph

Syntax hints:   F/wnf 1436   [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  nfsbv  1920  hbsb  1922  sbco2yz  1936  sbcomxyyz  1945  hbsbd  1957  nfsb4or  1998  sb8eu  2012  nfeu  2018  cbvab  2263  cbvralf  2648  cbvrexf  2649  cbvreu  2652  cbvralsv  2668  cbvrexsv  2669  cbvrab  2684  cbvreucsf  3064  cbvrabcsf  3065  cbvopab1  4001  cbvmptf  4022  cbvmpt  4023  ralxpf  4685  rexxpf  4686  cbviota  5093  sb8iota  5095  cbvriota  5740  dfoprab4f  6091