Theorem csbeq2d

Description: Formula-building deduction for class substitution. (Contributed by NM, 22-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypotheses	csbeq2d.1	⊢ Ⅎ 𝑥 𝜑
	csbeq2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	csbeq2d	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		csbeq2d.1	⊢ Ⅎ 𝑥 𝜑
2		csbeq2d.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	eleq2d	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
4	1 3	sbcbid	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 ↔ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 ) )
5	4	abbidv	⊢ ( 𝜑 → { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 } )
6		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 }
7		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 }
8	5 6 7	3eqtr4g	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )