Theorem dimatis 2172

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 2155 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 1560	. . . 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 1626	1 ⊢ ∃𝑥(𝜒 ∧ 𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371  ∃wex 1516

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)