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Metamath Proof Explorer

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 )