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 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐶 = ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ⦌ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		nfcv	⊢ Ⅎ 𝑥 𝐶
2	1	ax-gen	⊢ ∀ 𝑦 Ⅎ 𝑥 𝐶
3		csbnestgfw	⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑦 Ⅎ 𝑥 𝐶 ) → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐶 = ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ⦌ 𝐶 )
4	2 3	mpan2	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ⦋ 𝐵 / 𝑦 ⦌ 𝐶 = ⦋ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 / 𝑦 ⦌ 𝐶 )