Theorem nfsbcw

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 7-Sep-2014) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfsbcw.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfsbcw.2	$$⊢Ⅎx\phi$$
Assertion	nfsbcw	$$⊢Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$

Proof

Step	Hyp	Ref	Expression
1		nfsbcw.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfsbcw.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	nfsbcdw	$$⊢\top \to Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$
7	6	mptru	$$⊢Ⅎx\underset{\dot{}}{[}A/y\underset{\dot{}}{]}\phi$$