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Metamath Proof Explorer

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	$⊢ Ⅎ x φ$
	exlimdd.2	$⊢ Ⅎ x χ$
	exlimdd.3	$⊢ φ \to \exists x ψ$
	exlimimdd.4	$⊢ φ \to (ψ \to χ)$
Assertion	exlimimdd	$⊢ φ \to χ$

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		exlimimdd.4	$⊢ φ \to (ψ \to χ)$
5	1 2 4	exlimd	$⊢ φ \to (\exists x ψ \to χ)$
6	3 5	mpd	$⊢ φ \to χ$