Theorem sbco2vh 1964

Description: This is a version of sbco2 1984 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 1963	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1779	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1540	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1963	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1540	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1963	. 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 1362   [wsb 1776

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777

This theorem is referenced by:  nfsb  1965  equsb3  1970  sbn  1971  sbim  1972  sbor  1973  sban  1974  sbco2vd  1986  sbco3v  1988  sbcom2v2  2005  sbcom2  2006  dfsb7  2010  sb7f  2011  sbal  2019  sbal1  2021  sbex  2023