Theorem csbeq2dv

Description: Formula-building deduction for class substitution. (Contributed by NM, 10-Nov-2005) (Revised by Mario Carneiro, 1-Sep-2015)

	Ref	Expression
Hypothesis	csbeq2dv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
Assertion	csbeq2dv	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		csbeq2dv.1	⊢ ( 𝜑 → 𝐵 = 𝐶 )
2	1	eleq2d	⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐶 ) )
3	2	sbcbidv	⊢ ( 𝜑 → ( [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 ↔ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 ) )
4	3	abbidv	⊢ ( 𝜑 → { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 } = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 } )
5		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐵 }
6		df-csb	⊢ ⦋ 𝐴 / 𝑥 ⦌ 𝐶 = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝑦 ∈ 𝐶 }
7	4 5 6	3eqtr4g	⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ 𝐵 = ⦋ 𝐴 / 𝑥 ⦌ 𝐶 )