Theorem ferio 2304

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 2303	1 ⊢ ∃x(χ ∧ ¬ ψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is used by: (None)