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Theorem frege56a 43332
Description: Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege56a (((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))

Proof of Theorem frege56a
StepHypRef Expression
1 frege55cor1a 43330 . 2 ((𝜓𝜑) → (𝜑𝜓))
2 frege9 43273 . 2 (((𝜓𝜑) → (𝜑𝜓)) → (((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)))))
31, 2ax-mp 5 1 (((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  if-wif 1060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege28 43291  ax-frege52a 43318  ax-frege54a 43323
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061
This theorem is referenced by:  frege57a  43334
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