Theorem dimatis 2195

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 2178 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 1582	. . . 4 ⊢ (𝜓 → 𝜒)
4	3	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
5		simpl 109	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
6	4, 5	jca 306	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜑))
7	1, 6	eximii 1648	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1393  ∃wex 1538

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)