Theorem sbco2d 2416

Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
sbco2d.1	|- F/ x ph
sbco2d.2	|- F/ z ph
sbco2d.3	|- ( ph -> F/ 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 |- F/ z ph
2		sbco2d.3	. . . . 5 |- ( ph -> F/ z ps )
3	1, 2	nfim1 2067	. . . 4 |- F/ z ( ph -> ps )
4	3	sbco2 2415	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / x ] ( ph -> ps ) )
5		sbco2d.1	. . . . . 6 |- F/ x ph
6	5	sbrim 2396	. . . . 5 |- ( [ z / x ] ( ph -> ps ) <-> ( ph -> [ z / x ] ps ) )
7	6	sbbii 1887	. . . 4 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> [ y / z ] ( ph -> [ z / x ] ps ) )
8	1	sbrim 2396	. . . 4 |- ( [ y / z ] ( ph -> [ z / x ] ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
9	7, 8	bitri 264	. . 3 |- ( [ y / z ] [ z / x ] ( ph -> ps ) <-> ( ph -> [ y / z ] [ z / x ] ps ) )
10	5	sbrim 2396	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( ph -> [ y / x ] ps ) )
11	4, 9, 10	3bitr3i 290	. 2 |- ( ( ph -> [ y / z ] [ z / x ] ps ) <-> ( ph -> [ y / x ] ps ) )
12	11	pm5.74ri 261	1 |- ( ph -> ( [ y / z ] [ z / x ] ps <-> [ y / x ] ps ) )

Syntax hints:   -> wi 4   <-> wb 196   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco3  2417