Theorem nfsbv 1998

Description: If z is not free in ph, it is not free in [ y / x ] ph when z is distinct from x and y. Version of nfsb 1997 requiring more disjoint variables. (Contributed by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on x , y. (Revised by Steven Nguyen, 13-Aug-2023.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
nfsbv.nf	|- F/ z ph

Assertion

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

Distinct variable groups:   x, z   y, z

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

Proof of Theorem nfsbv

Step	Hyp	Ref	Expression
1		df-sb 1809	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2		nfv 1574	. . . 4 |- F/ z x = y
3		nfsbv.nf	. . . 4 |- F/ z ph
4	2, 3	nfim 1618	. . 3 |- F/ z ( x = y -> ph )
5	2, 3	nfan 1611	. . . 4 |- F/ z ( x = y /\ ph )
6	5	nfex 1683	. . 3 |- F/ z E. x ( x = y /\ ph )
7	4, 6	nfan 1611	. 2 |- F/ z ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) )
8	1, 7	nfxfr 1520	1 |- F/ z [ y / x ] ph

Syntax hints:   -> wi 4   /\ wa 104   F/wnf 1506   E.wex 1538   [wsb 1808

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbco2v  1999  cbvabw  2352