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) Avoid ax-13 . (Revised by Gino Giotto, 26-Jan-2024)

	Ref	Expression
Assertion	csbnestgw	$$⊢A\in V\to ⦋A/x⦌⦋B/y⦌C=⦋⦋A/x⦌B/y⦌C$$

Proof

Step	Hyp	Ref	Expression
1		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
2	1	ax-gen	$$⊢\forall y\underset{\_}{Ⅎ}xC$$
3		csbnestgfw	$$⊢\left(A\in V\wedge \forall y\underset{\_}{Ⅎ}xC\right)\to ⦋A/x⦌⦋B/y⦌C=⦋⦋A/x⦌B/y⦌C$$
4	2 3	mpan2	$$⊢A\in V\to ⦋A/x⦌⦋B/y⦌C=⦋⦋A/x⦌B/y⦌C$$