Theorem dariiALT 2659

Description: Alternate proof of darii 2658, shorter but using more axioms. This shows how the use of spi 2175 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2175 is the inference associated with sp 2174, which corresponds to the axiom (T) of modal logic. (Contributed by David A. Wheeler, 27-Aug-2016.) Added precisions on axiom usage. (Revised by BJ, 27-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
dariiALT	⊢ ∃𝑥(𝜒 ∧ 𝜓)

Proof of Theorem dariiALT

Step	Hyp	Ref	Expression
1		darii.min	. 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑)
2		darii.maj	. . . 4 ⊢ ∀𝑥(𝜑 → 𝜓)
3	2	spi 2175	. . 3 ⊢ (𝜑 → 𝜓)
4	3	anim2i 615	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1837	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-12 2169

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780

This theorem is referenced by: (None)