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Metamath Proof Explorer

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	$⊢ \underline{Ⅎ} x A$
	nfsbc.2	$⊢ Ⅎ x φ$
Assertion	nfsbc	$⊢ Ⅎ x \underset{˙}{[} A / y \underset{˙}{]} φ$

Proof

Step	Hyp	Ref	Expression
1		nfsbc.1	$⊢ \underline{Ⅎ} x A$
2		nfsbc.2	$⊢ Ⅎ x φ$
3		nftru	$⊢ Ⅎ y ⊤$
4	1	a1i	$⊢ ⊤ \to \underline{Ⅎ} x A$
5	2	a1i	$⊢ ⊤ \to Ⅎ x φ$
6	3 4 5	nfsbcd	$⊢ ⊤ \to Ⅎ x \underset{˙}{[} A / y \underset{˙}{]} φ$
7	6	mptru	$⊢ Ⅎ x \underset{˙}{[} A / y \underset{˙}{]} φ$