Theorem sbcom4

Description: Commutativity law for substitution. This theorem was incorrectly used as our previous version of pm11.07 but may still be useful. (Contributed by Andrew Salmon, 17-Jun-2011) (Proof shortened by Jim Kingdon, 22-Jan-2018)

	Ref	Expression
Assertion	sbcom4	|- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph )

Proof

Step	Hyp	Ref	Expression
1		sbv	|- ( [ w / x ] ph <-> ph )
2		sbv	|- ( [ y / z ] ph <-> ph )
3	2	sbbii	|- ( [ w / x ] [ y / z ] ph <-> [ w / x ] ph )
4		sbv	|- ( [ w / z ] ph <-> ph )
5	4	sbbii	|- ( [ y / x ] [ w / z ] ph <-> [ y / x ] ph )
6		sbv	|- ( [ y / x ] ph <-> ph )
7	5 6	bitri	|- ( [ y / x ] [ w / z ] ph <-> ph )
8	1 3 7	3bitr4i	|- ( [ w / x ] [ y / z ] ph <-> [ y / x ] [ w / z ] ph )