Theorem csbnestg

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker csbnestgw when possible. (Contributed by NM, 23-Nov-2005) (Proof shortened by Mario Carneiro, 10-Nov-2016) (New usage is discouraged.)

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

Proof

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