Theorem sbco2 2111

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 1936	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
2		sbequ 2067	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3	1, 2	bitr3d 255	. . 3 |- ( z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
4	3	sps 1800	. 2 |- ( A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
5		nfna1 1837	. . 3 |- F/ z -. A. z z = y
6		sbco2.1	. . . 4 |- F/ z ph
7	6	nfsb4 2081	. . 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 2102	. 2 |- ( -. A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
10	4, 9	pm2.61i 164	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   A.wal 1367   F/wnf 1589   [wsb 1700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701

This theorem is referenced by:  sbco2d  2113  equsb3ALT  2138  elsb3  2139  elsb4  2140  sb7f  2163  sbco4lem  2178  sbco4  2179  2eu6OLD  2372  eqsb3  2544  clelsb3  2545  sbralie  2960  sbcco  3209  clelsb3f  25864  bj-clelsb3  32359