Theorem exlimimdd

Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020) Shorten exlimdd . (Revised by Wolf Lammen, 3-Sep-2023)

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

Proof

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