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Theorem frege52aid 38654
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 38653 . 2 ((𝜑𝜓) → (if-(𝜑, ⊤, ⊥) → if-(𝜓, ⊤, ⊥)))
2 ifpid2 38317 . 2 (𝜑 ↔ if-(𝜑, ⊤, ⊥))
3 ifpid2 38317 . 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 1050  wtru 1633  wfal 1637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege52a 38653
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-tru 1635  df-fal 1638
This theorem is referenced by:  frege53aid  38655  frege57aid  38668  frege75  38734  frege89  38748  frege105  38764
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