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Theorem frege65a 43207
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2652 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege65a (((𝜓𝜒) ∧ (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Proof of Theorem frege65a
StepHypRef Expression
1 ifpimim 42833 . . 3 (if-(𝜑, (𝜓𝜒), (𝜏𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜒, 𝜂)))
2 frege64a 43206 . . 3 ((if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
31, 2syl 17 . 2 (if-(𝜑, (𝜓𝜒), (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
4 frege61a 43203 . 2 ((if-(𝜑, (𝜓𝜒), (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜓𝜒) ∧ (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))))
53, 4ax-mp 5 1 (((𝜓𝜒) ∧ (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  if-wif 1059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege58a 43199
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060
This theorem is referenced by:  frege66a  43208
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