Theorem bnj345 32593

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	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒))

Proof of Theorem bnj345

Step	Hyp	Ref	Expression
1		bnj334 32592	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃))
2		bnj250 32580	. . 3 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃) ↔ (𝜒 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜃)))
3		3anass 1093	. . 3 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜒 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜃)))
4	2, 3	bitr4i 277	. 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃) ↔ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃))
5		3anrev 1099	. . 3 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒))
6		bnj250 32580	. . . 4 ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
7		3anass 1093	. . . 4 ⊢ ((𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒)))
8	6, 7	bitr4i 277	. . 3 ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒))
9	5, 8	bitr4i 277	. 2 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒))
10	1, 4, 9	3bitri 296	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   ∧ w-bnj17 32565

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 206  df-an 396  df-3an 1087  df-bnj17 32566

This theorem is referenced by:  bnj422  32594  bnj446  32596  bnj929  32816  bnj964  32823