Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj250 Structured version   Visualization version   GIF version

Theorem bnj250 32147
 Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj250 ((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))

Proof of Theorem bnj250
StepHypRef Expression
1 df-bnj17 32133 . 2 ((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))
2 3anass 1092 . . 3 ((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
32anbi1i 626 . 2 (((𝜑𝜓𝜒) ∧ 𝜃) ↔ ((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃))
4 anass 472 . 2 (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))
51, 3, 43bitri 300 1 ((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∧ w-bnj17 32132 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-3an 1086  df-bnj17 32133 This theorem is referenced by:  bnj251  32148  bnj252  32149  bnj345  32160
 Copyright terms: Public domain W3C validator