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Mirrors > Home > MPE Home > Th. List > ccased | Structured version Visualization version GIF version |
Description: Deduction for combining cases. (Contributed by NM, 9-May-2004.) |
Ref | Expression |
---|---|
ccased.1 | ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂)) |
ccased.2 | ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜂)) |
ccased.3 | ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜂)) |
ccased.4 | ⊢ (𝜑 → ((𝜃 ∧ 𝜏) → 𝜂)) |
Ref | Expression |
---|---|
ccased | ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccased.1 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜂)) | |
2 | 1 | com12 32 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → (𝜑 → 𝜂)) |
3 | ccased.2 | . . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜂)) | |
4 | 3 | com12 32 | . . 3 ⊢ ((𝜃 ∧ 𝜒) → (𝜑 → 𝜂)) |
5 | ccased.3 | . . . 4 ⊢ (𝜑 → ((𝜓 ∧ 𝜏) → 𝜂)) | |
6 | 5 | com12 32 | . . 3 ⊢ ((𝜓 ∧ 𝜏) → (𝜑 → 𝜂)) |
7 | ccased.4 | . . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜏) → 𝜂)) | |
8 | 7 | com12 32 | . . 3 ⊢ ((𝜃 ∧ 𝜏) → (𝜑 → 𝜂)) |
9 | 2, 4, 6, 8 | ccase 1035 | . 2 ⊢ (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → (𝜑 → 𝜂)) |
10 | 9 | com12 32 | 1 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∧ (𝜒 ∨ 𝜏)) → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → 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: fpwwe2lem12 10398 mulge0 11493 zmulcl 12369 gcdabsOLD 16239 lcmabs 16310 pospo 18063 mulgass 18740 indistopon 22151 lgsdir2lem5 26477 outsideofeq 34432 smprngopr 36210 cdlemg33 38725 monotoddzzfi 40764 acongtr 40800 |
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