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Mirrors > Home > ILE Home > Th. List > fresison | GIF version |
Description: "Fresison", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓 (PeM), and some 𝜓 is 𝜒 (MiS), therefore some 𝜒 is not 𝜑 (SoP). (In Aristotelian notation, EIO-4: PeM and MiS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.) |
Ref | Expression |
---|---|
fresison.maj | ⊢ ∀𝑥(𝜑 → ¬ 𝜓) |
fresison.min | ⊢ ∃𝑥(𝜓 ∧ 𝜒) |
Ref | Expression |
---|---|
fresison | ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fresison.min | . 2 ⊢ ∃𝑥(𝜓 ∧ 𝜒) | |
2 | simpr 103 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
3 | fresison.maj | . . . . . 6 ⊢ ∀𝑥(𝜑 → ¬ 𝜓) | |
4 | 3 | spi 1429 | . . . . 5 ⊢ (𝜑 → ¬ 𝜓) |
5 | 4 | con2i 557 | . . . 4 ⊢ (𝜓 → ¬ 𝜑) |
6 | 5 | adantr 261 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → ¬ 𝜑) |
7 | 2, 6 | jca 290 | . 2 ⊢ ((𝜓 ∧ 𝜒) → (𝜒 ∧ ¬ 𝜑)) |
8 | 1, 7 | eximii 1493 | 1 ⊢ ∃𝑥(𝜒 ∧ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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