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Mirrors > Home > MPE Home > Th. List > 3bior1fd | Structured version Visualization version GIF version |
Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 936. (Contributed by Alexander van der Vekens, 8-Sep-2017.) |
Ref | Expression |
---|---|
3biorfd.1 | ⊢ (𝜑 → ¬ 𝜃) |
Ref | Expression |
---|---|
3bior1fd | ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3biorfd.1 | . . 3 ⊢ (𝜑 → ¬ 𝜃) | |
2 | biorf 936 | . . 3 ⊢ (¬ 𝜃 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓)))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓)))) |
4 | 3orass 1091 | . 2 ⊢ ((𝜃 ∨ 𝜒 ∨ 𝜓) ↔ (𝜃 ∨ (𝜒 ∨ 𝜓))) | |
5 | 3, 4 | bitr4di 292 | 1 ⊢ (𝜑 → ((𝜒 ∨ 𝜓) ↔ (𝜃 ∨ 𝜒 ∨ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 846 ∨ w3o 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 210 df-or 847 df-3or 1089 |
This theorem is referenced by: 3bior1fand 1477 3bior2fd 1478 nb3grprlem2 27323 |
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