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 205  wa 397  wo 846  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 206  df-an 398  df-or 847  df-ifp 1063
This theorem is referenced by:  dfifp3  1065  dfifp5  1067  ifpdfbi  1070  ifpnOLD  1074  ifpimpda  1082  revwlk  34115  ifpbi2  42218  ifpbi3  42219  ifpbi1  42228  ifpbi12  42239  ifpbi13  42240  ifpimimb  42255  ifpororb  42256  ifpbibib  42261  frege54cor0a  42614
  Copyright terms: Public domain W3C validator