Theorem ferio 2115

Description: "Ferio" ("Ferioque"), one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜑, therefore some 𝜒 is not 𝜓. (In Aristotelian notation, EIO-1: MeP and SiM therefore SoP.) For example, given "No homework is fun" and "Some reading is homework", therefore "Some reading is not fun". This is essentially a logical axiom in Aristotelian logic. Example from https://en.wikipedia.org/wiki/Syllogism. (Contributed by David A. Wheeler, 24-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
ferio	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem ferio

Step	Hyp	Ref	Expression
1		ferio.maj	. 2 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		ferio.min	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑)
3	1, 2	darii 2114	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)