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Theorem fesapo 2734
 Description: "Fesapo", one of the syllogisms of Aristotelian logic. No 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜓 exist, therefore some 𝜒 is not 𝜑. (In Aristotelian notation, EAO-4: PeM and MaS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)
Hypotheses
Ref Expression
fesapo.maj 𝑥(𝜑 → ¬ 𝜓)
fesapo.min 𝑥(𝜓𝜒)
fesapo.e 𝑥𝜓
Assertion
Ref Expression
fesapo 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem fesapo
StepHypRef Expression
1 fesapo.e . 2 𝑥𝜓
2 fesapo.min . . . 4 𝑥(𝜓𝜒)
32spi 2208 . . 3 (𝜓𝜒)
4 fesapo.maj . . . . 5 𝑥(𝜑 → ¬ 𝜓)
54spi 2208 . . . 4 (𝜑 → ¬ 𝜓)
65con2i 136 . . 3 (𝜓 → ¬ 𝜑)
73, 6jca 501 . 2 (𝜓 → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 1912 1 𝑥(𝜒 ∧ ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382  ∀wal 1629  ∃wex 1852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853 This theorem is referenced by: (None)
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