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Theorem bamalip 2021
 Description: "Bamalip", one of the syllogisms of Aristotelian logic. All 𝜑 is 𝜓, all 𝜓 is 𝜒, and 𝜑 exist, therefore some 𝜒 is 𝜑. (In Aristotelian notation, AAI-4: PaM and MaS therefore SiP.) Like barbari 2002. (Contributed by David A. Wheeler, 28-Aug-2016.)
Hypotheses
Ref Expression
bamalip.maj 𝑥(𝜑𝜓)
bamalip.min 𝑥(𝜓𝜒)
bamalip.e 𝑥𝜑
Assertion
Ref Expression
bamalip 𝑥(𝜒𝜑)

Proof of Theorem bamalip
StepHypRef Expression
1 bamalip.e . 2 𝑥𝜑
2 bamalip.maj . . . . 5 𝑥(𝜑𝜓)
32spi 1429 . . . 4 (𝜑𝜓)
4 bamalip.min . . . . 5 𝑥(𝜓𝜒)
54spi 1429 . . . 4 (𝜓𝜒)
63, 5syl 14 . . 3 (𝜑𝜒)
76ancri 307 . 2 (𝜑 → (𝜒𝜑))
81, 7eximii 1493 1 𝑥(𝜒𝜑)
 Colors of variables: wff set class Syntax hints:   → 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-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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