Theorem sbco2 2135

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

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		sbco2.1	. . . . . 6 |- F/ z ph
2	1	sbid2 2133	. . . . 5 |- ( [ x / z ] [ z / x ] ph <-> ph )
3		sbequ 2109	. . . . 5 |- ( x = y -> ( [ x / z ] [ z / x ] ph <-> [ y / z ] [ z / x ] ph ) )
4	2, 3	syl5bbr 251	. . . 4 |- ( x = y -> ( ph <-> [ y / z ] [ z / x ] ph ) )
5		sbequ12 1940	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6	4, 5	bitr3d 247	. . 3 |- ( x = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
7	6	sps 1766	. 2 |- ( A. x x = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
8		nfnae 2008	. . . 4 |- F/ x -. A. x x = y
9	1	nfs1 2093	. . . . 5 |- F/ x [ z / x ] ph
10	9	nfsb4 2130	. . . 4 |- ( -. A. x x = y -> F/ x [ y / z ] [ z / x ] ph )
11	4	a1i 11	. . . 4 |- ( -. A. x x = y -> ( x = y -> ( ph <-> [ y / z ] [ z / x ] ph ) ) )
12	8, 10, 11	sbied 2085	. . 3 |- ( -. A. x x = y -> ( [ y / x ] ph <-> [ y / z ] [ z / x ] ph ) )
13	12	bicomd 193	. 2 |- ( -. A. x x = y -> ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph ) )
14	7, 13	pm2.61i 158	1 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   A.wal 1546   F/wnf 1550   [wsb 1655

This theorem is referenced by:  sbco2d  2136  equsb3  2151  elsb3  2152  elsb4  2153  sb7f  2169  2eu6  2339  eqsb3  2505  clelsb3  2506  sbralie  2905  sbcco  3143  clelsb3f  23924

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656