Theorem exlimdd

Description: Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017) (Proof shortened by Wolf Lammen, 3-Sep-2023)

	Ref	Expression
Hypotheses	exlimdd.1	$$⊢Ⅎx\phi$$
	exlimdd.2	$$⊢Ⅎx\chi$$
	exlimdd.3	$$⊢\phi \to \exists x\psi$$
	exlimdd.4	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
Assertion	exlimdd	$$⊢\phi \to \chi$$

Proof

Step	Hyp	Ref	Expression
1		exlimdd.1	$$⊢Ⅎx\phi$$
2		exlimdd.2	$$⊢Ⅎx\chi$$
3		exlimdd.3	$$⊢\phi \to \exists x\psi$$
4		exlimdd.4	$$⊢\left(\phi \wedge \psi \right)\to \chi$$
5	4	ex	$$⊢\phi \to \left(\psi \to \chi \right)$$
6	1 2 3 5	exlimimdd	$$⊢\phi \to \chi$$