Theorem sbco2v 1941

Description: Version of sbco2 1958 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 1940	. 2 |- F/ z [ y / x ] ph
3		sbequ 1833	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
4	2, 3	sbiev 1785	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 104   F/wnf 1453   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by: (None)