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Theorem frege56a 40095
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 40093 . 2 ((𝜓𝜑) → (𝜑𝜓))
2 frege9 40036 . 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 207  if-wif 1054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 40014  ax-frege2 40015  ax-frege8 40033  ax-frege28 40054  ax-frege52a 40081  ax-frege54a 40086
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055
This theorem is referenced by:  frege57a  40097
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