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Mirrors > Home > MPE Home > Th. List > Mathboxes > notornotel1 | Structured version Visualization version GIF version |
Description: A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
Ref | Expression |
---|---|
notornotel1.1 | ⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒)) |
Ref | Expression |
---|---|
notornotel1 | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notornotel1.1 | . 2 ⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒)) | |
2 | ioran 984 | . . . 4 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ↔ (¬ ¬ 𝜓 ∧ ¬ 𝜒)) | |
3 | 2 | biimpi 219 | . . 3 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) → (¬ ¬ 𝜓 ∧ ¬ 𝜒)) |
4 | simpl 486 | . . 3 ⊢ ((¬ ¬ 𝜓 ∧ ¬ 𝜒) → ¬ ¬ 𝜓) | |
5 | notnotr 132 | . . 3 ⊢ (¬ ¬ 𝜓 → 𝜓) | |
6 | 3, 4, 5 | 3syl 18 | . 2 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) → 𝜓) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 |
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-an 400 df-or 848 |
This theorem is referenced by: notornotel2 35948 ac6s6 36024 |
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