Theorem csbie

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024)

	Ref	Expression
Hypotheses	csbie.1	⊢ 𝐴 ∈ V
	csbie.2	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
Assertion	csbie	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶

Proof

Step	Hyp	Ref	Expression
1		csbie.1	⊢ 𝐴 ∈ V
2		csbie.2	⊢ ( 𝑥 = 𝐴 → 𝐵 = 𝐶 )
3		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 }
4	2	eleq2d	⊢ ( 𝑥 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
5	1 4	sbcie	⊢ ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 )
6	5	abbii	⊢ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } = { 𝑦 ∣ 𝑦 ∈ 𝐶 }
7		abid2	⊢ { 𝑦 ∣ 𝑦 ∈ 𝐶 } = 𝐶
8	3 6 7	3eqtri	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = 𝐶