Theorem csbmpt2

Description: Move substitution into the second part of a maps-to notation. (Contributed by AV, 26-Sep-2019)

	Ref	Expression
Assertion	csbmpt2	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ( 𝑦 ∈ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		csbmpt12	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )
2		csbconstg	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ 𝑌 = 𝑌 )
3	2	mpteq1d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑦 ∈ ⦋ 𝐴 / 𝑥 ⦌ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) = ( 𝑦 ∈ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )
4	1 3	eqtrd	⊢ ( 𝐴 ∈ 𝑉 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝑦 ∈ 𝑌 ↦ 𝑍 ) = ( 𝑦 ∈ 𝑌 ↦ ⦋ 𝐴 / 𝑥 ⦌ 𝑍 ) )