Theorem darapti 2129

Description: "Darapti", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜑 is 𝜒, and some 𝜑 exist, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AAI-3: MaP and MaS therefore SiP.) For example, "All squares are rectangles" and "All squares are rhombuses", therefore "Some rhombuses are rectangles". (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

Ref	Expression
darapti.maj	⊢ ∀𝑥(𝜑 → 𝜓)
darapti.min	⊢ ∀𝑥(𝜑 → 𝜒)
darapti.e	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
darapti	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem darapti

Step	Hyp	Ref	Expression
1		darapti.e	. 2 ⊢ ∃𝑥𝜑
2		darapti.min	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜒)
3	2	spi 1524	. . 3 ⊢ (𝜑 → 𝜒)
4		darapti.maj	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜓)
5	4	spi 1524	. . 3 ⊢ (𝜑 → 𝜓)
6	3, 5	jca 304	. 2 ⊢ (𝜑 → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1590	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → 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-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)