Theorem baroco 2121

Description: "Baroco", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜒 is not 𝜓, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, AOO-2: PaM and SoM therefore SoP.) For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative." (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
baroco	⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem baroco

Step	Hyp	Ref	Expression
1		baroco.min	. 2 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜓)
2		baroco.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
3	2	spi 1524	. . . 4 ⊢ (𝜑 → 𝜓)
4	3	con3i 622	. . 3 ⊢ (¬ 𝜓 → ¬ 𝜑)
5	4	anim2i 340	. 2 ⊢ ((𝜒 ∧ ¬ 𝜓) → (𝜒 ∧ ¬ 𝜑))
6	1, 5	eximii 1590	1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)