Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege67a Structured version   Visualization version   GIF version

Theorem frege67a 39018
 Description: Lemma for frege68a 39019. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege67a ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))

Proof of Theorem frege67a
StepHypRef Expression
1 ax-frege58a 39008 . 2 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
2 frege7 38941 . 2 (((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒)) → ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))))
31, 2ax-mp 5 1 ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  if-wif 1089 This theorem was proved from axioms:  ax-mp 5  ax-frege1 38923  ax-frege2 38924  ax-frege58a 39008 This theorem is referenced by:  frege68a  39019
 Copyright terms: Public domain W3C validator