Theorem sbco2vlem 1861

Description: This is a version of sbco2 1880 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1862 which only requires that z and x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) (One distinct variable constraint removed by Jim Kingdon, 3-Feb-2018.)

Hypothesis

Ref	Expression
sbco2vlem.1	|- ( ph -> A. z ph )

Assertion

Ref	Expression
sbco2vlem	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Distinct variable groups:   x, z   y, z

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

Proof of Theorem sbco2vlem

Step	Hyp	Ref	Expression
1		sbco2vlem.1	. . 3 |- ( ph -> A. z ph )
2	1	hbsbv 1858	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
3		sbequ 1761	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
4	2, 3	sbieh 1713	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:  sbco2v  1862