Theorem alsi2d

Description: Deduction rule: Given "all some" applied to a top-level inference, you can extract the "exists" part. (Contributed by David A. Wheeler, 20-Oct-2018)

	Ref	Expression
Hypothesis	alsi2d.1	$$⊢\phi \to \forall !x\left(\psi \to \chi \right)$$
Assertion	alsi2d	$$⊢\phi \to \exists x\psi$$

Proof

Step	Hyp	Ref	Expression
1		alsi2d.1	$$⊢\phi \to \forall !x\left(\psi \to \chi \right)$$
2		df-alsi	$$⊢\forall !x\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\left(\psi \to \chi \right)\wedge \exists x\psi \right)$$
3	1 2	sylib	$$⊢\phi \to \left(\forall x\left(\psi \to \chi \right)\wedge \exists x\psi \right)$$
4	3	simprd	$$⊢\phi \to \exists x\psi$$