Theorem csbnestgw

Description: Nest the composition of two substitutions. Version of csbnestg with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 23-Nov-2005) (Revised by Gino Giotto, 26-Jan-2024)

	Ref	Expression
Assertion	csbnestgw	|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Proof

Step	Hyp	Ref	Expression
1		nfcv	|- F/_ x C
2	1	ax-gen	|- A. y F/_ x C
3		csbnestgfw	|- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
4	2 3	mpan2	|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )