Theorem sbco2vh 1945

Description: This is a version of sbco2 1965 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 1944	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1765	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1526	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1944	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1526	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1944	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	3, 5, 7	3bitr3i 210	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  nfsb  1946  equsb3  1951  sbn  1952  sbim  1953  sbor  1954  sban  1955  sbco2vd  1967  sbco3v  1969  sbcom2v2  1986  sbcom2  1987  dfsb7  1991  sb7f  1992  sbal  2000  sbal1  2002  sbex  2004