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	No typesetting found for |- ( ph -> AE x e. A ( ps -> ch ) ) with typecode |-
Assertion	rals2d	$$⊢\phi \to \exists x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		rals2d.1	Could not format ( ph -> AE x e. A ( ps -> ch ) ) : No typesetting found for |- ( ph -> AE x e. A ( ps -> ch ) ) with typecode |-
2		df-rals	Could not format ( AE x e. A ( ps -> ch ) <-> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) ) : No typesetting found for |- ( AE x e. A ( ps -> ch ) <-> ( A. x e. A ( ps -> ch ) /\ E. x e. A ps ) ) with typecode |-
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	$$⊢\phi \to \exists x\in A\psi$$