Theorem sbcocom 2030

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 1838	. . 3 |- [ z / y ] y = z
2		sbequ 1893	. . . 4 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sbimi 1817	. . 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 2019	. 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 1815

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816

This theorem is referenced by:  sbcomv  2031  sbco3xzyz  2033  sbcom  2035