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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-orel12 | Structured version Visualization version GIF version |
Description: In a conjunctive normal form a pair of nodes like (𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒) eliminates the need of a node (𝜓 ∨ 𝜒). This theorem allows simplifications in that respect. (Contributed by Wolf Lammen, 20-Jun-2020.) |
Ref | Expression |
---|---|
wl-orel12 | ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.1 894 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) | |
2 | orel1 886 | . . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓)) | |
3 | orc 864 | . . . 4 ⊢ (𝜓 → (𝜓 ∨ 𝜒)) | |
4 | 2, 3 | syl6com 37 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → (𝜓 ∨ 𝜒))) |
5 | notnot 142 | . . . . 5 ⊢ (𝜑 → ¬ ¬ 𝜑) | |
6 | orel1 886 | . . . . 5 ⊢ (¬ ¬ 𝜑 → ((¬ 𝜑 ∨ 𝜒) → 𝜒)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → ((¬ 𝜑 ∨ 𝜒) → 𝜒)) |
8 | olc 865 | . . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒)) | |
9 | 7, 8 | syl6com 37 | . . 3 ⊢ ((¬ 𝜑 ∨ 𝜒) → (𝜑 → (𝜓 ∨ 𝜒))) |
10 | 4, 9 | jaao 952 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → ((¬ 𝜑 ∨ 𝜑) → (𝜓 ∨ 𝜒))) |
11 | 1, 10 | mpi 20 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ (¬ 𝜑 ∨ 𝜒)) → (𝜓 ∨ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 |
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 206 df-an 397 df-or 845 |
This theorem is referenced by: wl-cases2-dnf 35680 |
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