Theorem sbco2 2415

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.)

Hypothesis

Ref	Expression
sbco2.1	|- F/ z ph

Assertion

Ref	Expression
sbco2	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Proof of Theorem sbco2

Step	Hyp	Ref	Expression
1		sbequ12 2111	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
2		sbequ 2376	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3	1, 2	bitr3d 270	. . 3 |- ( z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
4	3	sps 2055	. 2 |- ( A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
5		nfnae 2318	. . 3 |- F/ z -. A. z z = y
6		sbco2.1	. . . 4 |- F/ z ph
7	6	nfsb4 2390	. . 3 |- ( -. A. z z = y -> F/ z [ y / x ] ph )
8	2	a1i 11	. . 3 |- ( -. A. z z = y -> ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) ) )
9	5, 7, 8	sbied 2409	. 2 |- ( -. A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
10	4, 9	pm2.61i 176	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   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:  sbco2d  2416  equsb3ALT  2433  elsb3  2434  elsb4  2435  sb7f  2453  sbco4lem  2465  sbco4  2466  eqsb3  2728  clelsb3  2729  cbvab  2746  clelsb3f  2768  sbralie  3184  sbcco  3458  bj-clelsb3  32848