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Metamath Proof Explorer

Theorem datisi

Description: "Datisi", one of the syllogisms of Aristotelian logic. All ph is ps , and some ph is ch , therefore some ch is ps . In Aristotelian notation, AII-3: MaP and MiS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	datisi.maj	$⊢ \forall x (φ \to ψ)$
	datisi.min	$⊢ \exists x (φ \land χ)$
Assertion	datisi	$⊢ \exists x (χ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		datisi.maj	$⊢ \forall x (φ \to ψ)$
2		datisi.min	$⊢ \exists x (φ \land χ)$
3		exancom	$⊢ \exists x (φ \land χ) \leftrightarrow \exists x (χ \land φ)$
4	2 3	mpbi	$⊢ \exists x (χ \land φ)$
5	1 4	darii	$⊢ \exists x (χ \land ψ)$