Theorem sbco2vh 1998

Description: This is a version of sbco2 2018 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 1997	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1813	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1574	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1997	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1574	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1997	. 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 1395   [wsb 1810

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  nfsb  1999  equsb3  2004  sbn  2005  sbim  2006  sbor  2007  sban  2008  sbco2vd  2020  sbco3v  2022  sbcom2v2  2039  sbcom2  2040  dfsb7  2044  sb7f  2045  sbal  2053  sbal1  2055  sbex  2057