Theorem dimatis 2170

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 2153 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 1558	. . . 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 1624	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1370  ∃wex 1514

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)