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Theorem frege56a 37668
 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 37666 . 2 ((𝜓𝜑) → (𝜑𝜓))
2 frege9 37609 . 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 196  if-wif 1011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37587  ax-frege2 37588  ax-frege8 37606  ax-frege28 37627  ax-frege52a 37654  ax-frege54a 37659 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012 This theorem is referenced by:  frege57a  37670
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