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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj334 | Structured version Visualization version GIF version | ||
| Description: ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bnj334 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bnj290 34724 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜃 ∧ 𝜓)) | |
| 2 | bnj257 34721 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓 ∧ 𝜃)) | |
| 3 | bnj312 34726 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃)) | |
| 4 | 1, 2, 3 | 3bitri 297 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ w-bnj17 34700 | 
| 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 207 df-an 396 df-3an 1089 df-bnj17 34701 | 
| This theorem is referenced by: bnj345 34728 bnj518 34900 bnj916 34947 bnj929 34950 | 
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