Theorem nfsb 1975

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 1971	. . 3 |- F/ z [ w / x ] ph
3	2	nfsbxy 1971	. 2 |- F/ z [ y / w ] [ w / x ] ph
4		ax-17 1550	. . . 4 |- ( ph -> A. w ph )
5	4	sbco2vh 1974	. . 3 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
6	5	nfbii 1497	. 2 |- ( F/ z [ y / w ] [ w / x ] ph <-> F/ z [ y / x ] ph )
7	3, 6	mpbi 145	1 |- F/ z [ y / x ] ph

Syntax hints:   F/wnf 1484   [wsb 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  hbsb  1978  sbco2yz  1992  sbcomxyyz  2001  hbsbd  2011  nfsb4or  2050  sb8eu  2068  nfeu  2074  cbvab  2331  cbvralf  2733  cbvrexf  2734  cbvreu  2740  cbvralsv  2758  cbvrexsv  2759  cbvrab  2774  cbvreucsf  3166  cbvrabcsf  3167  cbvopab1  4133  cbvmptf  4154  cbvmpt  4155  ralxpf  4842  rexxpf  4843  cbviota  5256  sb8iota  5258  cbvriota  5933  dfoprab4f  6302