Theorem datisi 2051

Description: "Datisi", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, and some 𝜑 is 𝜒, therefore some 𝜒 is 𝜓. (In Aristotelian notation, AII-3: MaP and MiS therefore SiP.) (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
datisi	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem datisi

Step	Hyp	Ref	Expression
1		datisi.min	. 2 ⊢ ∃𝑥(𝜑 ∧ 𝜒)
2		simpr 108	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜒)
3		datisi.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
4	3	spi 1469	. . . 4 ⊢ (𝜑 → 𝜓)
5	4	adantr 270	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜓)
6	2, 5	jca 300	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1533	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  ferison  2053