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Theorem festino 2049
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.)
Hypotheses
Ref Expression
festino.maj 𝑥(𝜑 → ¬ 𝜓)
festino.min 𝑥(𝜒𝜓)
Assertion
Ref Expression
festino 𝑥(𝜒 ∧ ¬ 𝜑)

Proof of Theorem festino
StepHypRef Expression
1 festino.min . 2 𝑥(𝜒𝜓)
2 festino.maj . . . . 5 𝑥(𝜑 → ¬ 𝜓)
32spi 1470 . . . 4 (𝜑 → ¬ 𝜓)
43con2i 590 . . 3 (𝜓 → ¬ 𝜑)
54anim2i 334 . 2 ((𝜒𝜓) → (𝜒 ∧ ¬ 𝜑))
61, 5eximii 1534 1 𝑥(𝜒 ∧ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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