Theorem sbco2 2134

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 1961	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
2		sbequ 2090	. . . 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 1814	. 2 |- ( A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
5		nfna1 1851	. . 3 |- F/ z -. A. z z = y
6		sbco2.1	. . . 4 |- F/ z ph
7	6	nfsb4 2104	. . 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 2125	. 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 1377   F/wnf 1599   [wsb 1711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712

This theorem is referenced by:  sbco2d  2136  equsb3ALT  2160  elsb3  2161  elsb4  2162  sb7f  2185  sbco4lem  2200  sbco4  2201  2eu6OLD  2394  eqsb3  2587  clelsb3  2588  cbvab  2608  sbralie  3101  sbcco  3354  clelsb3f  27071  bj-clelsb3  33514