Theorem ralsd

Description: Introduction rule for "all some" restricted to a class. This is the converse of rals1d and rals2d taken together. (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypotheses	ralsd.1	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
	ralsd.2	$$⊢\phi \to \exists x\in A\psi$$
Assertion	ralsd	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ralsd.1	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
2		ralsd.2	$$⊢\phi \to \exists x\in A\psi$$
3		df-rals	$$⊢\forall \exists x\in A\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\in A\left(\psi \to \chi \right)\wedge \exists x\in A\psi \right)$$
4	1 2 3	sylanbrc	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$