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Metamath Proof Explorer

Theorem nfmodv

Description: Bound-variable hypothesis builder for the at-most-one quantifier. See nfmod for a version without disjoint variable conditions but requiring ax-13 . (Contributed by Mario Carneiro, 14-Nov-2016) (Revised by BJ, 28-Jan-2023)

	Ref	Expression
Hypotheses	nfmodv.1	$⊢ Ⅎ y φ$
	nfmodv.2	$⊢ φ \to Ⅎ x ψ$
Assertion	nfmodv	$⊢ φ \to Ⅎ x \exists^{*} y ψ$

Proof

Step	Hyp	Ref	Expression
1		nfmodv.1	$⊢ Ⅎ y φ$
2		nfmodv.2	$⊢ φ \to Ⅎ x ψ$
3		df-mo	$⊢ \exists^{*} y ψ \leftrightarrow \exists z \forall y (ψ \to y = z)$
4		nfv	$⊢ Ⅎ z φ$
5		nfvd	$⊢ φ \to Ⅎ x y = z$
6	2 5	nfimd	$⊢ φ \to Ⅎ x (ψ \to y = z)$
7	1 6	nfald	$⊢ φ \to Ⅎ x \forall y (ψ \to y = z)$
8	4 7	nfexd	$⊢ φ \to Ⅎ x \exists z \forall y (ψ \to y = z)$
9	3 8	nfxfrd	$⊢ φ \to Ⅎ x \exists^{*} y ψ$