Theorem nfsb 1819

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/zph

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/zph
2	1	nfsbxy 1815	. . 3 |- F/z[w / x]ph
3	2	nfsbxy 1815	. 2 |- F/z[y / w][w / x]ph
4		ax-17 1416	. . . 4 |- (ph -> A.wph)
5	4	sbco2v 1818	. . 3 |- ([y / w][w / x]ph <-> [y / x]ph)
6	5	nfbii 1359	. 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 1346  [wsb 1642

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  hbsb  1820  sbco2yz  1834  sbcomxyyz  1843  hbsbd  1855  nfsb4or  1896  sb8eu  1910  nfeu  1916  cbvab  2157  cbvralf  2521  cbvrexf  2522  cbvreu  2525  cbvralsv  2538  cbvrexsv  2539  cbvrab  2549  cbvreucsf  2904  cbvrabcsf  2905  cbvopab1  3821  cbvmpt  3842  ralxpf  4425  rexxpf  4426  cbviota  4815  sb8iota  4817  cbvriota  5421  dfoprab4f  5761