Theorem nfcsbd

Description: Deduction version of nfcsb . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 21-Nov-2005) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfcsbd.1	⊢ Ⅎ 𝑦 𝜑
	nfcsbd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
	nfcsbd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion	nfcsbd	⊢ ( 𝜑 → Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		nfcsbd.1	⊢ Ⅎ 𝑦 𝜑
2		nfcsbd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
3		nfcsbd.3	⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
4		df-csb	⊢ ⦋ 𝐴 / 𝑦 ⦌ 𝐵 = { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 }
5		nfv	⊢ Ⅎ 𝑧 𝜑
6	3	nfcrd	⊢ ( 𝜑 → Ⅎ 𝑥 𝑧 ∈ 𝐵 )
7	1 2 6	nfsbcd	⊢ ( 𝜑 → Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 )
8	5 7	nfabd	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 } )
9	4 8	nfcxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵 )