Theorem sbcnestgw

Description: Nest the composition of two substitutions. Version of sbcnestg with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 27-Nov-2005) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

	Ref	Expression
Assertion	sbcnestgw	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}⦋A/x⦌B/y\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfv	$$⊢Ⅎx\phi$$
2	1	ax-gen	$$⊢\forall yℲx\phi$$
3		sbcnestgfw	$$⊢\left(A\in V\wedge \forall yℲx\phi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}⦋A/x⦌B/y\underset{\dot{}}{]}\phi \right)$$
4	2 3	mpan2	$$⊢A\in V\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\underset{\dot{}}{[}B/y\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}⦋A/x⦌B/y\underset{\dot{}}{]}\phi \right)$$