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Metamath Proof Explorer

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	$⊢ φ \to \exists x ψ$
	exlimdd.4	$⊢ (φ \land ψ) \to χ$
Assertion	exlimdd	$⊢ φ \to χ$

Proof

Step	Hyp	Ref	Expression
1		exlimdd.1	$⊢ Ⅎ x φ$
2		exlimdd.2	$⊢ Ⅎ x χ$
3		exlimdd.3	$⊢ φ \to \exists x ψ$
4		exlimdd.4	$⊢ (φ \land ψ) \to χ$
5	4	ex	$⊢ φ \to (ψ \to χ)$
6	1 2 3 5	exlimimdd	$⊢ φ \to χ$