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Theorem frege57aid 40570
Description: This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 231. 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 40556 . 2 ((𝜓𝜑) → (𝜓𝜑))
2 frege56aid 40568 . 2 (((𝜓𝜑) → (𝜓𝜑)) → ((𝜑𝜓) → (𝜓𝜑)))
31, 2ax-mp 5 1 ((𝜑𝜓) → (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 40488  ax-frege2 40489  ax-frege8 40507  ax-frege52a 40555
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-fal 1551
This theorem is referenced by:  frege68a  40584  frege68b  40611  frege68c  40629  frege100  40661
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