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Theorem frege65a 37694
 Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2562 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 37370 . . 3 (if-(𝜑, (𝜓𝜒), (𝜏𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜒, 𝜂)))
2 frege64a 37693 . . 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 37690 . 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 384  if-wif 1011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37601  ax-frege2 37602  ax-frege8 37620  ax-frege58a 37686 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012 This theorem is referenced by:  frege66a  37695
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