Theorem sbco3 1948

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 1947	. . 3 |- ( [ w / y ] [ y / x ] ph <-> [ w / x ] [ x / y ] ph )
2	1	sbbii 1739	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / w ] [ w / x ] [ x / y ] ph )
3		ax-17 1507	. . 3 |- ( [ y / x ] ph -> A. w [ y / x ] ph )
4	3	sbco2h 1938	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
5		ax-17 1507	. . 3 |- ( [ x / y ] ph -> A. w [ x / y ] ph )
6	5	sbco2h 1938	. 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 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737

This theorem is referenced by:  sbcom  1949