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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnf1dd | Structured version Visualization version GIF version |
Description: A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
Ref | Expression |
---|---|
cnf1dd.1 | ⊢ (𝜑 → (𝜓 → ¬ 𝜒)) |
cnf1dd.2 | ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) |
Ref | Expression |
---|---|
cnf1dd | ⊢ (𝜑 → (𝜓 → 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnf1dd.1 | . . 3 ⊢ (𝜑 → (𝜓 → ¬ 𝜒)) | |
2 | cnf1dd.2 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) | |
3 | 1, 2 | jcad 513 | . 2 ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∧ (𝜒 ∨ 𝜃)))) |
4 | df-or 845 | . . 3 ⊢ ((𝜒 ∨ 𝜃) ↔ (¬ 𝜒 → 𝜃)) | |
5 | pm3.35 800 | . . 3 ⊢ ((¬ 𝜒 ∧ (¬ 𝜒 → 𝜃)) → 𝜃) | |
6 | 4, 5 | sylan2b 594 | . 2 ⊢ ((¬ 𝜒 ∧ (𝜒 ∨ 𝜃)) → 𝜃) |
7 | 3, 6 | syl6 35 | 1 ⊢ (𝜑 → (𝜓 → 𝜃)) |
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: cnf2dd 36243 cnfn1dd 36244 mpobi123f 36314 mptbi12f 36318 ac6s6 36324 |
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