Theorem festino 2675

Description: "Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.)

Hypotheses

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

Assertion

Ref	Expression
festino	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem festino

Step	Hyp	Ref	Expression
1		festino.maj	. . 3 ⊢ ∀𝑥(𝜑 → ¬ 𝜓)
2		con2 135	. . . . 5 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑))
3	2	anim2d 612	. . . 4 ⊢ ((𝜑 → ¬ 𝜓) → ((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
4	3	alimi 1814	. . 3 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑)))
5	1, 4	ax-mp 5	. 2 ⊢ ∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑))
6		festino.min	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜓)
7		exim 1836	. 2 ⊢ (∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒 ∧ 𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑)))
8	5, 6, 7	mp2 9	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783

This theorem is referenced by:  fresison  2690