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Theorem frege64a 37693
Description: Lemma for frege65a 37694. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege64a ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))

Proof of Theorem frege64a
StepHypRef Expression
1 frege62a 37691 . 2 (if-(𝜎, 𝜒, 𝜂) → (((𝜒𝜃) ∧ (𝜂𝜁)) → if-(𝜎, 𝜃, 𝜁)))
2 frege18 37629 . 2 ((if-(𝜎, 𝜒, 𝜂) → (((𝜒𝜃) ∧ (𝜂𝜁)) → if-(𝜎, 𝜃, 𝜁))) → ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))))
31, 2ax-mp 5 1 ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 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:  frege65a  37694
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