Theorem fesapo 2691

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

Hypotheses

Ref	Expression
fesapo.maj	⊢ ∀𝑥(𝜑 → ¬ 𝜓)
fesapo.min	⊢ ∀𝑥(𝜓 → 𝜒)
fesapo.e	⊢ ∃𝑥𝜓

Assertion

Ref	Expression
fesapo	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fesapo

Step	Hyp	Ref	Expression
1		fesapo.maj	. . 3 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		con2 135	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
3	2	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥(𝜓 → ¬ 𝜑))
4	1, 3	ax-mp 5	. 2 ⊢ ∀𝑥(𝜓 → ¬ 𝜑)
5		fesapo.min	. 2 ⊢ ∀𝑥(𝜓 → 𝜒)
6		fesapo.e	. 2 ⊢ ∃𝑥𝜓
7	4, 5, 6	felapton 2686	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)