This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nfabd

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfabdw when possible. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-9 and ax-ext . (Revised by Wolf Lammen, 23-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfabd.1	$⊢ Ⅎ y φ$
	nfabd.2	$⊢ φ \to Ⅎ x ψ$
Assertion	nfabd	$⊢ φ \to \underline{Ⅎ} x \{y \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		nfabd.1	$⊢ Ⅎ y φ$
2		nfabd.2	$⊢ φ \to Ⅎ x ψ$
3		nfv	$⊢ Ⅎ z φ$
4		df-clab	$⊢ z \in \{y \| ψ\} \leftrightarrow [z/ y] ψ$
5	1 2	nfsbd	$⊢ φ \to Ⅎ x [z/ y] ψ$
6	4 5	nfxfrd	$⊢ φ \to Ⅎ x z \in \{y \| ψ\}$
7	3 6	nfcd	$⊢ φ \to \underline{Ⅎ} x \{y \| ψ\}$