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	⊢ Ⅎ 𝑥 𝜑
	exlimdd.2	⊢ Ⅎ 𝑥 𝜒
	exlimdd.3	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
	exlimdd.4	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
Assertion	exlimdd	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		exlimdd.1	⊢ Ⅎ 𝑥 𝜑
2		exlimdd.2	⊢ Ⅎ 𝑥 𝜒
3		exlimdd.3	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
4		exlimdd.4	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
5	4	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
6	1 2 3 5	exlimimdd	⊢ ( 𝜑 → 𝜒 )