Theorem sbco2v 1862

Description: This is a version of sbco2 1880 where z is distinct from x. (Contributed by Jim Kingdon, 12-Feb-2018.)

Hypothesis

Ref	Expression
sbco2v.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
sbco2v	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Distinct variable group:   x, z

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

Proof of Theorem sbco2v

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco2v.1	. . . 4 |- ( ph -> A. z ph )
2	1	sbco2vlem 1861	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1688	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1459	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1861	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1459	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1861	. 2 |- ( [ y / w ] [ w / x ] ph <-> [ y / x ] ph )
8	3, 5, 7	3bitr3i 208	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  nfsb  1863  equsb3  1866  sbn  1867  sbim  1868  sbor  1869  sban  1870  sbco2vd  1882  sbco3v  1884  sbcom2v2  1903  sbcom2  1904  dfsb7  1908  sb7f  1909  sbal  1917  sbal1  1919  sbex  1921