Theorem ralsn0d

Description: Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypothesis	ralsn0d.1	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$
Assertion	ralsn0d	$$⊢\phi \to A\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ralsn0d.1	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$
2	1	rals2d	$$⊢\phi \to \exists x\in A\psi$$
3		rexn0	$$⊢\exists x\in A\psi \to A\ne \varnothing$$
4	2 3	syl	$$⊢\phi \to A\ne \varnothing$$