Theorem sbcocom 1999

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 1809	. . 3 |- [ z / y ] y = z
2		sbequ 1864	. . . 4 |- ( y = z -> ( [ y / x ] ph <-> [ z / x ] ph ) )
3	2	sbimi 1788	. . 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 1988	. 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 1786

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787

This theorem is referenced by:  sbcomv  2000  sbco3xzyz  2002  sbcom  2004