Theorem dimatis 2017

Description: "Dimatis", one of the syllogisms of Aristotelian logic. Some 𝜑 is 𝜓, and all 𝜓 is 𝜒, therefore some 𝜒 is 𝜑. (In Aristotelian notation, IAI-4: PiM and MaS therefore SiP.) For example, "Some pets are rabbits.", "All rabbits have fur", therefore "Some fur bearing animals are pets". Like darii 2000 with positions interchanged. (Contributed by David A. Wheeler, 28-Aug-2016.)

Hypotheses

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

Assertion

Ref	Expression
dimatis	⊢ ∃𝑥(𝜒 ∧ 𝜑)

Proof of Theorem dimatis

Step	Hyp	Ref	Expression
1		dimatis.maj	. 2 ⊢ ∃𝑥(𝜑 ∧ 𝜓)
2		dimatis.min	. . . . 5 ⊢ ∀𝑥(𝜓 → 𝜒)
3	2	spi 1429	. . . 4 ⊢ (𝜓 → 𝜒)
4	3	adantl 262	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
5		simpl 102	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
6	4, 5	jca 290	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜑))
7	1, 6	eximii 1493	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)