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Theorem fresison 2144
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.)
Hypotheses
Ref Expression
fresison.maj 𝑥(𝜑 → ¬ 𝜓)
fresison.min 𝑥(𝜓𝜒)
Assertion
Ref Expression
fresison 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fresison
StepHypRef Expression
1 fresison.min . 2 𝑥(𝜓𝜒)
2 simpr 110 . . 3 ((𝜓𝜒) → 𝜒)
3 fresison.maj . . . . . 6 𝑥(𝜑 → ¬ 𝜓)
43spi 1536 . . . . 5 (𝜑 → ¬ 𝜓)
54con2i 627 . . . 4 (𝜓 → ¬ 𝜑)
65adantr 276 . . 3 ((𝜓𝜒) → ¬ 𝜑)
72, 6jca 306 . 2 ((𝜓𝜒) → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1602 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1351  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534
This theorem depends on definitions:  df-bi 117
This theorem is referenced by: (None)
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