Theorem rals2d

Description: Deduction rule: Given "all some" applied to a class, you can extract the "there exists" part. Note that the witness must satisfy the antecedent ps , not merely be a member of A . (Contributed by David A. Wheeler, 20-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypothesis	rals2d.1	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$
Assertion	rals2d	$$⊢\phi \to \exists x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		rals2d.1	$$⊢\phi \to \forall \exists x\in A\left(\psi \to \chi \right)$$
2		df-rals	$$⊢\forall \exists x\in A\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\in A\left(\psi \to \chi \right)\wedge \exists x\in A\psi \right)$$
3	1 2	sylib	$$⊢\phi \to \left(\forall x\in A\left(\psi \to \chi \right)\wedge \exists x\in A\psi \right)$$
4	3	simprd	$$⊢\phi \to \exists x\in A\psi$$