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Theorem frege52aid 37661
Description: The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 205. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege52aid ((𝜑𝜓) → (𝜑𝜓))

Proof of Theorem frege52aid
StepHypRef Expression
1 ax-frege52a 37660 . 2 ((𝜑𝜓) → (if-(𝜑, ⊤, ⊥) → if-(𝜓, ⊤, ⊥)))
2 ifpid2 37323 . 2 (𝜑 ↔ if-(𝜑, ⊤, ⊥))
3 ifpid2 37323 . 2 (𝜓 ↔ if-(𝜓, ⊤, ⊥))
41, 2, 33imtr4g 285 1 ((𝜑𝜓) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  if-wif 1011  wtru 1481  wfal 1485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  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:  frege53aid  37662  frege57aid  37675  frege75  37741  frege89  37755  frege105  37771
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