Theorem dariiALT 2693

Description: Alternate proof of darii 2692, shorter but using more axioms. This shows how the use of spi 2220 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2220 is the inference associated with sp 2219, 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 2220	. . 3 ⊢ (𝜑 → 𝜓)
4	3	anim2i 628	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1867	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by: (None)