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	$$⊢\underset{\_}{Ⅎ}xA$$
	nfsbc.2	$$⊢Ⅎx\phi$$
Assertion	nfsbc	$$⊢Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfsbc.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfsbc.2	$$⊢Ⅎx\phi$$
3		nftru	$$⊢Ⅎy\top$$
4	1	a1i	$$⊢\top \to \underset{\_}{Ⅎ}xA$$
5	2	a1i	$$⊢\top \to Ⅎx\phi$$
6	3 4 5	nfsbcd	$$⊢\top \to Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$
7	6	mptru	$$⊢Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$