Theorem fresison 2689

Description: "Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). In Aristotelian notation, EIO-4: PeM and MiS therefore SoP. (Contributed by David A. Wheeler, 28-Aug-2016.) Shorten and reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
fresison	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison

Step	Hyp	Ref	Expression
1		fresison.maj	. 2 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		fresison.min	. . 3 ⊢ ∃𝑥(𝜓 ∧ 𝜒)
3		exancom 1861	. . 3 ⊢ (∃𝑥(𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜒 ∧ 𝜓))
4	2, 3	mpbi 230	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜓)
5	1, 4	festino 2674	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by: (None)