Theorem sbco2v 1876

Description: This is a version of sbco2 1894 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 1875	. . 3 |- ( [ w / z ] [ z / x ] ph <-> [ w / x ] ph )
3	2	sbbii 1702	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / w ] [ w / x ] ph )
4		ax-17 1471	. . 3 |- ( [ z / x ] ph -> A. w [ z / x ] ph )
5	4	sbco2vlem 1875	. 2 |- ( [ y / w ] [ w / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph )
6		ax-17 1471	. . 3 |- ( ph -> A. w ph )
7	6	sbco2vlem 1875	. 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 1294   [wsb 1699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700

This theorem is referenced by:  nfsb  1877  equsb3  1880  sbn  1881  sbim  1882  sbor  1883  sban  1884  sbco2vd  1896  sbco3v  1898  sbcom2v2  1917  sbcom2  1918  dfsb7  1922  sb7f  1923  sbal  1931  sbal1  1933  sbex  1935