Theorem dimatis 2159

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 2142 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 1547	. . . 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 1613	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

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: (None)