Theorem sbcom

Description: A commutativity law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . Check out sbcom3vv for a version requiring fewer axioms. (Contributed by NM, 27-May-1997) (Proof shortened by Wolf Lammen, 20-Sep-2018) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		sbco3	|- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcom3	|- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcom3	|- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1 2 3	3bitr3i	|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )