Theorem sbco3 1903

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 1902	. . 3 |- ( [ w / y ] [ y / x ] ph <-> [ w / x ] [ x / y ] ph )
2	1	sbbii 1702	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / w ] [ w / x ] [ x / y ] ph )
3		ax-17 1471	. . 3 |- ( [ y / x ] ph -> A. w [ y / x ] ph )
4	3	sbco2h 1893	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
5		ax-17 1471	. . 3 |- ( [ x / y ] ph -> A. w [ x / y ] ph )
6	5	sbco2h 1893	. 2 |- ( [ z / w ] [ w / x ] [ x / y ] ph <-> [ z / x ] [ x / y ] ph )
7	2, 4, 6	3bitr3i 209	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / x ] [ x / y ] ph )

Syntax hints:   <-> wb 104   [wsb 1699

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700

This theorem is referenced by:  sbcom  1904