Theorem nfsbc1d

Description: Deduction version of nfsbc1 . (Contributed by NM, 23-May-2006) (Revised by Mario Carneiro, 12-Oct-2016)

	Ref	Expression
Hypothesis	nfsbc1d.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
Assertion	nfsbc1d	⊢ ( 𝜑 → Ⅎ 𝑥 [ 𝐴 / 𝑥 ] 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfsbc1d.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )
2		df-sbc	⊢ ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
3		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜓 }
4	3	a1i	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑥 ∣ 𝜓 } )
5	1 4	nfeld	⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 ∈ { 𝑥 ∣ 𝜓 } )
6	2 5	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 [ 𝐴 / 𝑥 ] 𝜓 )