Theorem csbid

Description: Analogue of sbid for proper substitution into a class. (Contributed by NM, 10-Nov-2005)

	Ref	Expression
Assertion	csbid	⊢ ⦋ 𝑥 / 𝑥 ⦌ 𝐴 = 𝐴

Proof

Step	Hyp	Ref	Expression
1		df-csb	⊢ ⦋ 𝑥 / 𝑥 ⦌ 𝐴 = { 𝑦 ∣ [ 𝑥 / 𝑥 ] 𝑦 ∈ 𝐴 }
2		sbcid	⊢ ( [ 𝑥 / 𝑥 ] 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 )
3	2	abbii	⊢ { 𝑦 ∣ [ 𝑥 / 𝑥 ] 𝑦 ∈ 𝐴 } = { 𝑦 ∣ 𝑦 ∈ 𝐴 }
4		abid2	⊢ { 𝑦 ∣ 𝑦 ∈ 𝐴 } = 𝐴
5	1 3 4	3eqtri	⊢ ⦋ 𝑥 / 𝑥 ⦌ 𝐴 = 𝐴