ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  anandi GIF version

Theorem anandi 580
Description: Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
Assertion
Ref Expression
anandi ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Proof of Theorem anandi
StepHypRef Expression
1 anidm 394 . . 3 ((𝜑𝜑) ↔ 𝜑)
21anbi1i 454 . 2 (((𝜑𝜑) ∧ (𝜓𝜒)) ↔ (𝜑 ∧ (𝜓𝜒)))
3 an4 576 . 2 (((𝜑𝜑) ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
42, 3bitr3i 185 1 ((𝜑 ∧ (𝜓𝜒)) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  anandi3  976  moanim  2080  difundi  3359  inrab  3379  uniin  3792  xpcom  5131  fin  5355  fndmin  5573  nnaord  6453  ixpin  6665  ltexprlemdisj  7521  bldisj  12788  blininf  12811
  Copyright terms: Public domain W3C validator