Theorem sbcocom 1893

Description: Relationship between composition and commutativity for substitution. (Contributed by Jim Kingdon, 28-Feb-2018.)

Assertion

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

Proof of Theorem sbcocom

Step	Hyp	Ref	Expression
1		equsb1 1716	. . 3 |- [ z / y ] y = z
2		sbequ 1769	. . . 4 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sbimi 1695	. . 3 |- ( [ z / y ] y = z -> [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph ) )
4	1, 3	ax-mp 7	. 2 |- [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph )
5		sbbi 1882	. 2 |- ( [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph ) <-> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
6	4, 5	mpbi 144	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )

Syntax hints:   <-> wb 104   [wsb 1693

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 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694

This theorem is referenced by:  sbcomv  1894  sbco3xzyz  1896  sbcom  1898