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Theorem frege57a 43197
Description: Analogue of frege57aid 43196. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege57a ((𝜑𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))

Proof of Theorem frege57a
StepHypRef Expression
1 ax-frege52a 43181 . 2 ((𝜓𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
2 frege56a 43195 . 2 (((𝜓𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) → ((𝜑𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))))
31, 2ax-mp 5 1 ((𝜑𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  if-wif 1059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege28 43154  ax-frege52a 43181  ax-frege54a 43186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060
This theorem is referenced by: (None)
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