Theorem dariiALT 2751

Description: Alternate proof of darii 2750, shorter but using more axioms. This shows how the use of spi 2183 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2183 is the inference associated with sp 2182, 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 2183	. . 3 ⊢ (𝜑 → 𝜓)
4	3	anim2i 618	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1837	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781

This theorem is referenced by: (None)