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Mirrors > Home > MPE Home > Th. List > festino | Structured version Visualization version GIF version |
Description: "Festino", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, and some 𝜒 is 𝜓, therefore some 𝜒 is not 𝜑. In Aristotelian notation, EIO-2: PeM and SiM therefore SoP. (Contributed by David A. Wheeler, 25-Nov-2016.) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022.) |
Ref | Expression |
---|---|
festino.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
festino.min | ⊢ ∃𝑥(𝜒 ∧ 𝜓) |
Ref | Expression |
---|---|
festino | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | festino.maj | . . 3 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
2 | con2 137 | . . . . 5 ⊢ ((𝜑 → ¬ 𝜓) → (𝜓 → ¬ 𝜑)) | |
3 | 2 | anim2d 615 | . . . 4 ⊢ ((𝜑 → ¬ 𝜓) → ((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑))) |
4 | 3 | alimi 1819 | . . 3 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) → ∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑))) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ ∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑)) |
6 | festino.min | . 2 ⊢ ∃𝑥(𝜒 ∧ 𝜓) | |
7 | exim 1841 | . 2 ⊢ (∀𝑥((𝜒 ∧ 𝜓) → (𝜒 ∧ ¬ 𝜑)) → (∃𝑥(𝜒 ∧ 𝜓) → ∃𝑥(𝜒 ∧ ¬ 𝜑))) | |
8 | 5, 6, 7 | mp2 9 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1541 ∃wex 1787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: fresison 2690 |
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