Theorem sbco3 1889

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Assertion

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

Proof of Theorem sbco3

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbco3xzyz 1888	. . 3 |- ( [ w / y ] [ y / x ] ph <-> [ w / x ] [ x / y ] ph )
2	1	sbbii 1688	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / w ] [ w / x ] [ x / y ] ph )
3		ax-17 1459	. . 3 |- ( [ y / x ] ph -> A. w [ y / x ] ph )
4	3	sbco2h 1879	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
5		ax-17 1459	. . 3 |- ( [ x / y ] ph -> A. w [ x / y ] ph )
6	5	sbco2h 1879	. 2 |- ( [ z / w ] [ w / x ] [ x / y ] ph <-> [ z / x ] [ x / y ] ph )
7	2, 4, 6	3bitr3i 208	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbcom  1890