Theorem bocardo 2712

Description: "Bocardo", one of the syllogisms of Aristotelian logic. Some 𝜑 is not 𝜓, and all 𝜑 is 𝜒, therefore some 𝜒 is not 𝜓. Instance of disamis 2710. In Aristotelian notation, OAO-3: MoP and MaS therefore SoP. For example, "Some cats have no tails", "All cats are mammals", therefore "Some mammals have no tails". (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
bocardo	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

Proof of Theorem bocardo

Step	Hyp	Ref	Expression
1		bocardo.maj	. 2 ⊢ ∃𝑥(𝜑 ∧ ¬ 𝜓)
2		bocardo.min	. 2 ⊢ ∀𝑥(𝜑 → 𝜒)
3	1, 2	disamis 2710	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)