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Theorem fesapo 2020
 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 1429 . . 3 (𝜓𝜒)
4 fesapo.maj . . . . 5 𝑥(𝜑 → ¬ 𝜓)
54spi 1429 . . . 4 (𝜑 → ¬ 𝜓)
65con2i 557 . . 3 (𝜓 → ¬ 𝜑)
73, 6jca 290 . 2 (𝜓 → (𝜒 ∧ ¬ 𝜑))
81, 7eximii 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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