Theorem sbco2 2159

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 1993	. . . 4 |- ( z = y -> ( [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
2		sbequ 2118	. . . 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 1866	. 2 |- ( A. z z = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
5		nfnae 2059	. . 3 |- F/ z -. A. z z = y
6		sbco2.1	. . . 4 |- F/ z ph
7	6	nfsb4 2132	. . 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 2152	. 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 1393   F/wnf 1617   [wsb 1740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741

This theorem is referenced by:  sbco2d  2160  equsb3ALT  2178  elsb3  2179  elsb4  2180  sb7f  2198  sbco4lem  2210  sbco4  2211  2eu6OLD  2384  eqsb3  2577  clelsb3  2578  cbvab  2598  sbralie  3097  sbcco  3350  clelsb3f  27506  bj-clelsb3  34546  frege72  38085