Theorem ferison 2157

Description: "Ferison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-3: MeP and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
ferison	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem ferison

Step	Hyp	Ref	Expression
1		ferison.maj	. 2 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		ferison.min	. 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒)
3	1, 2	datisi 2155	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)