Theorem csbidm

Description: Idempotent law for class substitutions. (Contributed by NM, 1-Mar-2008) (Revised by NM, 18-Aug-2018)

	Ref	Expression
Assertion	csbidm	|- [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B

Proof

Step	Hyp	Ref	Expression
1		csbnest1g	|- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ [_ A / x ]_ A / x ]_ B )
2		csbconstg	|- ( A e. _V -> [_ A / x ]_ A = A )
3	2	csbeq1d	|- ( A e. _V -> [_ [_ A / x ]_ A / x ]_ B = [_ A / x ]_ B )
4	1 3	eqtrd	|- ( A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
5		csbprc	|- ( -. A e. _V -> [_ A / x ]_ [_ A / x ]_ B = (/) )
6		csbprc	|- ( -. A e. _V -> [_ A / x ]_ B = (/) )
7	5 6	eqtr4d	|- ( -. A e. _V -> [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B )
8	4 7	pm2.61i	|- [_ A / x ]_ [_ A / x ]_ B = [_ A / x ]_ B