MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfifp2 Structured version   Visualization version   GIF version

Theorem dfifp2 1064
Description: Alternate definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒". This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1063). (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
dfifp2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Proof of Theorem dfifp2
StepHypRef Expression
1 df-ifp 1063 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 cases2 1047 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
31, 2bitri 275 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  if-wif 1062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063
This theorem is referenced by:  dfifp3  1065  dfifp5  1067  ifpdfbi  1070  ifpimpda  1080  revwlk  35109  ifpbi2  43457  ifpbi3  43458  ifpbi1  43467  ifpbi12  43478  ifpbi13  43479  ifpimimb  43494  ifpororb  43495  ifpbibib  43500  frege54cor0a  43853
  Copyright terms: Public domain W3C validator