Theorem csbnest1g

Description: Nest the composition of two substitutions. (Contributed by NM, 23-May-2006) (Proof shortened by Mario Carneiro, 11-Nov-2016)

	Ref	Expression
Assertion	csbnest1g	$$⊢A\in V\to ⦋A/x⦌⦋B/x⦌C=⦋⦋A/x⦌B/x⦌C$$

Proof

Step	Hyp	Ref	Expression
1		nfcsb1v	$$⊢\underset{\_}{Ⅎ}x⦋y/x⦌C$$
2	1	ax-gen	$$⊢\forall y\underset{\_}{Ⅎ}x⦋y/x⦌C$$
3		csbnestgfw	$$⊢\left(A\in V\wedge \forall y\underset{\_}{Ⅎ}x⦋y/x⦌C\right)\to ⦋A/x⦌⦋B/y⦌⦋y/x⦌C=⦋⦋A/x⦌B/y⦌⦋y/x⦌C$$
4	2 3	mpan2	$$⊢A\in V\to ⦋A/x⦌⦋B/y⦌⦋y/x⦌C=⦋⦋A/x⦌B/y⦌⦋y/x⦌C$$
5		csbcow	$$⊢⦋B/y⦌⦋y/x⦌C=⦋B/x⦌C$$
6	5	csbeq2i	$$⊢⦋A/x⦌⦋B/y⦌⦋y/x⦌C=⦋A/x⦌⦋B/x⦌C$$
7		csbcow	$$⊢⦋⦋A/x⦌B/y⦌⦋y/x⦌C=⦋⦋A/x⦌B/x⦌C$$
8	4 6 7	3eqtr3g	$$⊢A\in V\to ⦋A/x⦌⦋B/x⦌C=⦋⦋A/x⦌B/x⦌C$$