Theorem sbco2vd 1965

Description: Version of sbco2d 1964 with a distinct variable constraint between x and z. (Contributed by Jim Kingdon, 19-Feb-2018.)

Hypotheses

Ref	Expression
sbco2vd.1	|- ( ph -> A. x ph )
sbco2vd.2	|- ( ph -> A. z ph )
sbco2vd.3	|- ( ph -> ( ps -> A. z ps ) )

Assertion

Ref	Expression
sbco2vd	|- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Distinct variable group:   x, z

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

Proof of Theorem sbco2vd

Step	Hyp	Ref	Expression
1		sbco2vd.2	. . . . 5 |- ( ph -> A. z ph )
2		sbco2vd.3	. . . . 5 |- ( ph -> ( ps -> A. z ps ) )
3	1, 2	hbim1 1568	. . . 4 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
4	3	sbco2vh 1943	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
5		sbco2vd.1	. . . . . 6 |- ( ph -> A. x ph )
6	5	sbrim 1954	. . . . 5 |- ( [ z / x ] ( ph -> ps ) <-> ( ph -> [ z / x ] ps ) )
7	6	sbbii 1763	. . . 4 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( ph -> [ z / x ] ps ) )
8	1	sbrim 1954	. . . 4 |- ( [ y / z ] ( ph -> [ z / x ] ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
9	7, 8	bitri 184	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
10	5	sbrim 1954	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
11	4, 9, 10	3bitr3i 210	. 2 |- ( ( ph -> [ y / z ] [ z / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
12	11	pm5.74ri 181	1 |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   [wsb 1760

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761

This theorem is referenced by: (None)