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Theorem frege56a 40223
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 40221 . 2 ((𝜓𝜑) → (𝜑𝜓))
2 frege9 40164 . 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 208  if-wif 1057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 40142  ax-frege2 40143  ax-frege8 40161  ax-frege28 40182  ax-frege52a 40209  ax-frege54a 40214
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058
This theorem is referenced by:  frege57a  40225
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