Theorem sbco2vlem 1820

Description: This is a version of sbco2 1839 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1821 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 1817	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
3		sbequ 1721	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
4	2, 3	sbieh 1673	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbco2v  1821