Theorem sbco2vlem 1963

Description: This is a version of sbco2 1984 where z is distinct from x and from y. It is a lemma on the way to proving sbco2v 1967 which only requires that z and x be distinct. (Contributed by Jim Kingdon, 25-Dec-2017.) Remove one disjoint variable condition. (Revised 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 1960	. 2 |- ( [ y / x ] ph -> A. z [ y / x ] ph )
3		sbequ 1854	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
4	2, 3	sbieh 1804	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:  sbco2vh  1964