MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfifp7 Structured version   Visualization version   GIF version

Theorem dfifp7 1065
Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
Assertion
Ref Expression
dfifp7 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))

Proof of Theorem dfifp7
StepHypRef Expression
1 orcom 867 . 2 (((𝜑𝜓) ∨ ¬ (𝜒𝜑)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
2 dfifp6 1064 . 2 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))
3 imor 850 . 2 (((𝜒𝜑) → (𝜑𝜓)) ↔ (¬ (𝜒𝜑) ∨ (𝜑𝜓)))
41, 2, 33bitr4i 306 1 (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))
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:  wl-2mintru2  34901
  Copyright terms: Public domain W3C validator