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| Mirrors > Home > MPE Home > Th. List > 3ioran | Structured version Visualization version GIF version | ||
| Description: Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.) |
| Ref | Expression |
|---|---|
| 3ioran | ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ioran 986 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) | |
| 2 | 1 | anbi1i 624 | . 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒)) |
| 3 | ioran 986 | . . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒)) | |
| 4 | df-3or 1088 | . . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) | |
| 5 | 3, 4 | xchnxbir 333 | . 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒)) |
| 6 | df-3an 1089 | . 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒)) | |
| 7 | 2, 5, 6 | 3bitr4i 303 | 1 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 |
| 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-or 849 df-3or 1088 df-3an 1089 |
| This theorem is referenced by: 3oran 1109 cadnot 1615 lcmftp 16673 prm23ge5 16853 cnfldfun 21378 cnfldfunOLD 21391 fbunfip 23877 frgrregord013 30414 wl-nfeqfb 37537 sn-inelr 42497 usgrexmpl2trifr 47996 gpg5nbgrvtx03starlem1 48024 gpg5nbgrvtx03starlem2 48025 gpg5nbgrvtx03starlem3 48026 gpg5nbgrvtx13starlem1 48027 gpg5nbgrvtx13starlem2 48028 gpg5nbgrvtx13starlem3 48029 |
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