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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj345 | 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 |
---|---|
bnj345 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj334 31983 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃)) | |
2 | bnj250 31971 | . . 3 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃) ↔ (𝜒 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜃))) | |
3 | 3anass 1091 | . . 3 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜒 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜃))) | |
4 | 2, 3 | bitr4i 280 | . 2 ⊢ ((𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃) ↔ (𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃)) |
5 | 3anrev 1097 | . . 3 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒)) | |
6 | bnj250 31971 | . . . 4 ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒))) | |
7 | 3anass 1091 | . . . 4 ⊢ ((𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (𝜃 ∧ ((𝜑 ∧ 𝜓) ∧ 𝜒))) | |
8 | 6, 7 | bitr4i 280 | . . 3 ⊢ ((𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜃 ∧ (𝜑 ∧ 𝜓) ∧ 𝜒)) |
9 | 5, 8 | bitr4i 280 | . 2 ⊢ ((𝜒 ∧ (𝜑 ∧ 𝜓) ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒)) |
10 | 1, 4, 9 | 3bitri 299 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∧ w-bnj17 31956 |
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 209 df-an 399 df-3an 1085 df-bnj17 31957 |
This theorem is referenced by: bnj422 31985 bnj446 31987 bnj929 32208 bnj964 32215 |
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