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Theorem frege57aid 37675
 Description: This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 218. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege57aid ((𝜑𝜓) → (𝜓𝜑))

Proof of Theorem frege57aid
StepHypRef Expression
1 frege52aid 37661 . 2 ((𝜓𝜑) → (𝜓𝜑))
2 frege56aid 37673 . 2 (((𝜓𝜑) → (𝜓𝜑)) → ((𝜑𝜓) → (𝜓𝜑)))
31, 2ax-mp 5 1 ((𝜑𝜓) → (𝜓𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37593  ax-frege2 37594  ax-frege8 37612  ax-frege52a 37660 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-tru 1483  df-fal 1486 This theorem is referenced by:  frege68a  37689  frege68b  37716  frege68c  37734  frege100  37766
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