Theorem nfsb 1864

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 1860	. . 3 |- F/ z [ w / x ] ph
3	2	nfsbxy 1860	. 2 |- F/ z [ y / w ] [ w / x ] ph
4		ax-17 1460	. . . 4 |- ( ph -> A. w ph )
5	4	sbco2v 1863	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
6	5	nfbii 1403	. 2 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
7	3, 6	mpbi 143	1 |- F/ z [ y / x ] ph

Syntax hints:   F/wnf 1390   [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  hbsb  1865  sbco2yz  1879  sbcomxyyz  1888  hbsbd  1900  nfsb4or  1941  sb8eu  1955  nfeu  1961  cbvab  2202  cbvralf  2572  cbvrexf  2573  cbvreu  2576  cbvralsv  2589  cbvrexsv  2590  cbvrab  2600  cbvreucsf  2967  cbvrabcsf  2968  cbvopab1  3859  cbvmpt  3880  ralxpf  4510  rexxpf  4511  cbviota  4902  sb8iota  4904  cbvriota  5509  dfoprab4f  5850