Theorem datisi 2152

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 110	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜒)
3		datisi.maj	. . . . 5 ⊢ ∀𝑥(𝜑 → 𝜓)
4	3	spi 1547	. . . 4 ⊢ (𝜑 → 𝜓)
5	4	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜓)
6	2, 5	jca 306	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜒 ∧ 𝜓))
7	1, 6	eximii 1613	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ferison  2154