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Mirrors > Home > MPE Home > Th. List > dariiALT | Structured version Visualization version GIF version |
Description: Alternate proof of darii 2663, shorter but using more axioms. This shows how the use of spi 2182 may shorten some proofs of the Aristotelian syllogisms, even though this adds axiom dependencies. Note that spi 2182 is the inference associated with sp 2181, 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.) |
Ref | Expression |
---|---|
darii.maj | ⊢ ∀𝑥(𝜑 → 𝜓) |
darii.min | ⊢ ∃𝑥(𝜒 ∧ 𝜑) |
Ref | Expression |
---|---|
dariiALT | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | darii.min | . 2 ⊢ ∃𝑥(𝜒 ∧ 𝜑) | |
2 | darii.maj | . . . 4 ⊢ ∀𝑥(𝜑 → 𝜓) | |
3 | 2 | spi 2182 | . . 3 ⊢ (𝜑 → 𝜓) |
4 | 3 | anim2i 617 | . 2 ⊢ ((𝜒 ∧ 𝜑) → (𝜒 ∧ 𝜓)) |
5 | 1, 4 | eximii 1834 | 1 ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 |
This theorem is referenced by: (None) |
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