Theorem sbcocom 1970

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 1785	. . 3 |- [ z / y ] y = z
2		sbequ 1840	. . . 4 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sbimi 1764	. . 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 1959	. 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 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbcomv  1971  sbco3xzyz  1973  sbcom  1975