Theorem nfrals

Description: Bound-variable hypothesis builder for "all some" restricted to a class. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	nfrals.1	$$⊢\underset{\_}{Ⅎ}xA$$
	nfrals.2	$$⊢Ⅎx\phi$$
	nfrals.3	$$⊢Ⅎx\psi$$
Assertion	nfrals	$$⊢Ⅎx\forall \exists y\in A\left(\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfrals.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfrals.2	$$⊢Ⅎx\phi$$
3		nfrals.3	$$⊢Ⅎx\psi$$
4		df-rals	$$⊢\forall \exists y\in A\left(\phi \to \psi \right)\leftrightarrow \left(\forall y\in A\left(\phi \to \psi \right)\wedge \exists y\in A\phi \right)$$
5	2 3	nfim	$$⊢Ⅎx\left(\phi \to \psi \right)$$
6	1 5	nfralw	$$⊢Ⅎx\forall y\in A\left(\phi \to \psi \right)$$
7	1 2	nfrexw	$$⊢Ⅎx\exists y\in A\phi$$
8	6 7	nfan	$$⊢Ⅎx\left(\forall y\in A\left(\phi \to \psi \right)\wedge \exists y\in A\phi \right)$$
9	4 8	nfxfr	$$⊢Ⅎx\forall \exists y\in A\left(\phi \to \psi \right)$$