Theorem sbco3 2417

Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.)

Assertion

Ref	Expression
sbco3	|- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Proof of Theorem sbco3

Step	Hyp	Ref	Expression
1		drsb1 2377	. . 3 |- ( A. x x = y -> ( [ z / x ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph ) )
2		nfae 2316	. . . 4 |- F/ x A. x x = y
3		sbequ12a 2113	. . . . 5 |- ( x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
4	3	sps 2055	. . . 4 |- ( A. x x = y -> ( [ y / x ] ph <-> [ x / y ] ph ) )
5	2, 4	sbbid 2403	. . 3 |- ( A. x x = y -> ( [ z / x ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
6	1, 5	bitr3d 270	. 2 |- ( A. x x = y -> ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
7		sbco 2412	. . . 4 |- ( [ x / y ] [ y / x ] ph <-> [ x / y ] ph )
8	7	sbbii 1887	. . 3 |- ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )
9		nfnae 2318	. . . 4 |- F/ y -. A. x x = y
10		nfnae 2318	. . . 4 |- F/ x -. A. x x = y
11		nfsb2 2360	. . . 4 |- ( -. A. x x = y -> F/ x [ y / x ] ph )
12	9, 10, 11	sbco2d 2416	. . 3 |- ( -. A. x x = y -> ( [ z / x ] [ x / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph ) )
13	8, 12	syl5rbbr 275	. 2 |- ( -. A. x x = y -> ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph ) )
14	6, 13	pm2.61i 176	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbcom  2418