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Metamath Proof Explorer

Theorem disamis

Description: "Disamis", one of the syllogisms of Aristotelian logic. Some ph is ps , and all ph is ch , therefore some ch is ps . In Aristotelian notation, IAI-3: MiP and MaS therefore SiP. (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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