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Metamath Proof Explorer

Theorem fresison

Description: "Fresison", one of the syllogisms of Aristotelian logic. No ph is ps (PeM), and some ps is ch (MiS), therefore some ch is not ph (SoP). In Aristotelian notation, EIO-4: PeM and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

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

Proof

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