Theorem festino

Description: "Festino", one of the syllogisms of Aristotelian logic. No ph is ps , and some ch is ps , therefore some ch is not ph . In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	festino.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
	festino.min	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )
Assertion	festino	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		festino.maj	⊢ ∀ 𝑥 ( 𝜑 → ¬ 𝜓 )
2		festino.min	⊢ ∃ 𝑥 ( 𝜒 ∧ 𝜓 )
3		con2	⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( 𝜓 → ¬ 𝜑 ) )
4	3	anim2d	⊢ ( ( 𝜑 → ¬ 𝜓 ) → ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) )
5	4	alimi	⊢ ( ∀ 𝑥 ( 𝜑 → ¬ 𝜓 ) → ∀ 𝑥 ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) )
6	1 5	ax-mp	⊢ ∀ 𝑥 ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) )
7		exim	⊢ ( ∀ 𝑥 ( ( 𝜒 ∧ 𝜓 ) → ( 𝜒 ∧ ¬ 𝜑 ) ) → ( ∃ 𝑥 ( 𝜒 ∧ 𝜓 ) → ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 ) ) )
8	6 2 7	mp2	⊢ ∃ 𝑥 ( 𝜒 ∧ ¬ 𝜑 )