Theorem sbco2d 1954

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sbco2d.1	|- ( ph -> A. x ph )
sbco2d.2	|- ( ph -> A. z ph )
sbco2d.3	|- ( ph -> ( ps -> A. z ps ) )

Assertion

Ref	Expression
sbco2d	|- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Proof of Theorem sbco2d

Step	Hyp	Ref	Expression
1		sbco2d.2	. . . . 5 |- ( ph -> A. z ph )
2		sbco2d.3	. . . . 5 |- ( ph -> ( ps -> A. z ps ) )
3	1, 2	hbim1 1558	. . . 4 |- ( ( ph -> ps ) -> A. z ( ph -> ps ) )
4	3	sbco2h 1952	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
5		sbco2d.1	. . . . . 6 |- ( ph -> A. x ph )
6	5	sbrim 1944	. . . . 5 |- ( [ z / x ] ( ph -> ps ) <-> ( ph -> [ z / x ] ps ) )
7	6	sbbii 1753	. . . 4 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( ph -> [ z / x ] ps ) )
8	1	sbrim 1944	. . . 4 |- ( [ y / z ] ( ph -> [ z / x ] ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
9	7, 8	bitri 183	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
10	5	sbrim 1944	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
11	4, 9, 10	3bitr3i 209	. 2 |- ( ( ph -> [ y / z ] [ z / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
12	11	pm5.74ri 180	1 |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)