Theorem dariiALT 2690

Description: Alternate proof of darii 2689, 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 629	. 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓))
5	1, 4	eximii 1870	1 ⊢ ∃𝑥(𝜒 ∧ 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)