MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfifp4 Structured version   Visualization version   GIF version

Theorem dfifp4 1062
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
Assertion
Ref Expression
dfifp4 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem dfifp4
StepHypRef Expression
1 dfifp3 1061 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
2 imor 850 . . 3 ((𝜑𝜓) ↔ (¬ 𝜑𝜓))
32anbi1i 626 . 2 (((𝜑𝜓) ∧ (𝜑𝜒)) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
41, 3bitri 278 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  if-wif 1058
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 210  df-an 400  df-or 845  df-ifp 1059
This theorem is referenced by:  anifp  1067  ifpan123g  40154  ifpan23  40155  ifpdfor2  40156  ifpdfor  40160  ifpim1  40164  ifpnot  40165  ifpid2  40166  ifpim2  40167  ifpnot23  40173  ifpidg  40186  ifpim123g  40195  ifpimim  40204
  Copyright terms: Public domain W3C validator