Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege52aid Structured version   Visualization version   GIF version

Theorem frege52aid 37172
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 204. 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 37171 . 2 ((𝜑𝜓) → (if-(𝜑, ⊤, ⊥) → if-(𝜓, ⊤, ⊥)))
2 ifpid2 36834 . 2 (𝜑 ↔ if-(𝜑, ⊤, ⊥))
3 ifpid2 36834 . 2 (𝜓 ↔ if-(𝜓, ⊤, ⊥))
41, 2, 33imtr4g 284 1 ((𝜑𝜓) → (𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  if-wif 1006  wtru 1476  wfal 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege52a 37171
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-tru 1478  df-fal 1481
This theorem is referenced by:  frege53aid  37173  frege57aid  37186  frege75  37252  frege89  37266  frege105  37282
  Copyright terms: Public domain W3C validator