Theorem sbco2 2220

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 2057	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
2		sbequ 2181	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3	1, 2	bitr3d 258	. . 3 |- ( z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
4	3	sps 1920	. 2 |- ( A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
5		nfnae 2124	. . 3 |- F/ z -. A. z z = y
6		sbco2.1	. . . 4 |- F/ z ph
7	6	nfsb4 2195	. . 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 2214	. 2 |- ( -. A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
10	4, 9	pm2.61i 167	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   A.wal 1435   F/wnf 1661   [wsb 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791

This theorem is referenced by:  sbco2d  2221  equsb3ALT  2239  elsb3  2240  elsb4  2241  sb7f  2259  sbco4lem  2271  sbco4  2272  eqsb3  2533  clelsb3  2534  cbvab  2551  sbralie  3009  sbcco  3265  clelsb3f  28058  bj-clelsb3  31368