Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege68a Structured version   Visualization version   GIF version

Theorem frege68a 37200
 Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege68a (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))

Proof of Theorem frege68a
StepHypRef Expression
1 frege57aid 37186 . 2 (((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒)))
2 frege67a 37199 . 2 ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
31, 2ax-mp 5 1 (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  if-wif 1006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege52a 37171  ax-frege58a 37189 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: (None)
 Copyright terms: Public domain W3C validator