Theorem nfsb 1822

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 1818	. . 3 |- F/ z [ w / x ] ph
3	2	nfsbxy 1818	. 2 |- F/ z [ y / w ] [ w / x ] ph
4		ax-17 1419	. . . 4 |- ( ph -> A. w ph )
5	4	sbco2v 1821	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
6	5	nfbii 1362	. 2 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
7	3, 6	mpbi 133	1 |- F/ z [ y / x ] ph

Syntax hints:   F/wnf 1349   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  hbsb  1823  sbco2yz  1837  sbcomxyyz  1846  hbsbd  1858  nfsb4or  1899  sb8eu  1913  nfeu  1919  cbvab  2160  cbvralf  2527  cbvrexf  2528  cbvreu  2531  cbvralsv  2544  cbvrexsv  2545  cbvrab  2555  cbvreucsf  2910  cbvrabcsf  2911  cbvopab1  3830  cbvmpt  3851  ralxpf  4482  rexxpf  4483  cbviota  4872  sb8iota  4874  cbvriota  5478  dfoprab4f  5819