Theorem sbcocom 2023

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 1833	. . 3 |- [ z / y ] y = z
2		sbequ 1888	. . . 4 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sbimi 1812	. . 3 |- ( [ z / y ] y = z -> [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph ) )
4	1, 3	ax-mp 5	. 2 |- [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph )
5		sbbi 2012	. 2 |- ( [ z / y ] ( [ y / x ] ph <-> [ z / x ] ph ) <-> ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph ) )
6	4, 5	mpbi 145	1 |- ( [ z / y ] [ y / x ] ph <-> [ z / y ] [ z / x ] ph )

Syntax hints:   <-> wb 105   [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sbcomv  2024  sbco3xzyz  2026  sbcom  2028