Theorem sbco3 1967

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 1966	. . 3 |- ( [ w / y ] [ y / x ] ph <-> [ w / x ] [ x / y ] ph )
2	1	sbbii 1758	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / w ] [ w / x ] [ x / y ] ph )
3		ax-17 1519	. . 3 |- ( [ y / x ] ph -> A. w [ y / x ] ph )
4	3	sbco2h 1957	. 2 |- ( [ z / w ] [ w / y ] [ y / x ] ph <-> [ z / y ] [ y / x ] ph )
5		ax-17 1519	. . 3 |- ( [ x / y ] ph -> A. w [ x / y ] ph )
6	5	sbco2h 1957	. 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 1755

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbcom  1968