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Theorem bnj345 31889
 Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj345 ((𝜑𝜓𝜒𝜃) ↔ (𝜃𝜑𝜓𝜒))

Proof of Theorem bnj345
StepHypRef Expression
1 bnj334 31888 . 2 ((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜑𝜓𝜃))
2 bnj250 31876 . . 3 ((𝜒𝜑𝜓𝜃) ↔ (𝜒 ∧ ((𝜑𝜓) ∧ 𝜃)))
3 3anass 1089 . . 3 ((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ↔ (𝜒 ∧ ((𝜑𝜓) ∧ 𝜃)))
42, 3bitr4i 279 . 2 ((𝜒𝜑𝜓𝜃) ↔ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃))
5 3anrev 1095 . . 3 ((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ↔ (𝜃 ∧ (𝜑𝜓) ∧ 𝜒))
6 bnj250 31876 . . . 4 ((𝜃𝜑𝜓𝜒) ↔ (𝜃 ∧ ((𝜑𝜓) ∧ 𝜒)))
7 3anass 1089 . . . 4 ((𝜃 ∧ (𝜑𝜓) ∧ 𝜒) ↔ (𝜃 ∧ ((𝜑𝜓) ∧ 𝜒)))
86, 7bitr4i 279 . . 3 ((𝜃𝜑𝜓𝜒) ↔ (𝜃 ∧ (𝜑𝜓) ∧ 𝜒))
95, 8bitr4i 279 . 2 ((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ↔ (𝜃𝜑𝜓𝜒))
101, 4, 93bitri 298 1 ((𝜑𝜓𝜒𝜃) ↔ (𝜃𝜑𝜓𝜒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   ∧ w-bnj17 31861 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 208  df-an 397  df-3an 1083  df-bnj17 31862 This theorem is referenced by:  bnj422  31890  bnj446  31892  bnj929  32113  bnj964  32120
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