Theorem nfsbc

Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfsbcw when possible. (Contributed by NM, 7-Sep-2014) (Revised by Mario Carneiro, 12-Oct-2016) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfsbc.1	⊢ Ⅎ 𝑥 𝐴
	nfsbc.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfsbc	⊢ Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfsbc.1	⊢ Ⅎ 𝑥 𝐴
2		nfsbc.2	⊢ Ⅎ 𝑥 𝜑
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
6	3 4 5	nfsbcd	⊢ ( ⊤ → Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑 )
7	6	mptru	⊢ Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑