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Mirrors > Home > ILE Home > Th. List > 3ioran | 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 747 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) | |
2 | 1 | anbi1i 455 | . 2 ⊢ ((¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒)) |
3 | ioran 747 | . . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒)) | |
4 | df-3or 974 | . . 3 ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) | |
5 | 3, 4 | xchnxbir 676 | . 2 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ (𝜑 ∨ 𝜓) ∧ ¬ 𝜒)) |
6 | df-3an 975 | . 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒)) | |
7 | 2, 5, 6 | 3bitr4i 211 | 1 ⊢ (¬ (𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 703 ∨ w3o 972 ∧ w3a 973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 |
This theorem is referenced by: ne3anior 2428 onntri35 7214 |
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