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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	⊢ ∃ 𝑥 ( 𝜑 ∧ 𝜓 )
	disamis.min	⊢ ∀ 𝑥 ( 𝜑 → 𝜒 )
Assertion	disamis	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		disamis.maj	⊢ ∃ 𝑥 ( 𝜑 ∧ 𝜓 )
2		disamis.min	⊢ ∀ 𝑥 ( 𝜑 → 𝜒 )
3	2 1	datisi	⊢ ∃ 𝑥 ( 𝜓 ∧ 𝜒 )
4		exancom	⊢ ( ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) ↔ ∃ 𝑥 ( 𝜒 ∧ 𝜓 ) )
5	3 4	mpbi	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )