Theorem sbco2v 1999

Description: Version of sbco2 2016 with disjoint variable conditions. (Contributed by Wolf Lammen, 29-Apr-2023.)

Hypothesis

Ref	Expression
sbco2v.1	|- F/ z ph

Assertion

Ref	Expression
sbco2v	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Distinct variable groups:   x, z   y, z

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

Proof of Theorem sbco2v

Step	Hyp	Ref	Expression
1		sbco2v.1	. . 3 |- F/ z ph
2	1	nfsbv 1998	. 2 |- F/ z [ y / x ] ph
3		sbequ 1886	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
4	2, 3	sbiev 1838	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 105   F/wnf 1506   [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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  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: (None)