Theorem sbco2vh 1938

Description: This is a version of sbco2 1958 where z is distinct from x. (Contributed by Jim Kingdon, 12-Feb-2018.)

Hypothesis

Ref	Expression
sbco2vh.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
sbco2vh	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Distinct variable group:   x, z

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

Proof of Theorem sbco2vh

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2vh.1	. . . 4 |- ( ph -> A. z ph )
2	1	sbco2vlem 1937	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1758	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1519	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1937	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1519	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1937	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	3, 5, 7	3bitr3i 209	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   [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:  nfsb  1939  equsb3  1944  sbn  1945  sbim  1946  sbor  1947  sban  1948  sbco2vd  1960  sbco3v  1962  sbcom2v2  1979  sbcom2  1980  dfsb7  1984  sb7f  1985  sbal  1993  sbal1  1995  sbex  1997