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Mirrors > Home > MPE Home > Th. List > ccase | Structured version Visualization version GIF version |
Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.) |
Ref | Expression |
---|---|
ccase.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜏) |
ccase.2 | ⊢ ((𝜒 ∧ 𝜓) → 𝜏) |
ccase.3 | ⊢ ((𝜑 ∧ 𝜃) → 𝜏) |
ccase.4 | ⊢ ((𝜒 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
ccase | ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccase.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏) | |
2 | ccase.2 | . . 3 ⊢ ((𝜒 ∧ 𝜓) → 𝜏) | |
3 | 1, 2 | jaoian 820 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) → 𝜏) |
4 | ccase.3 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏) | |
5 | ccase.4 | . . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏) | |
6 | 4, 5 | jaoian 820 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜃) → 𝜏) |
7 | 3, 6 | jaodan 822 | 1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 |
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 196 df-or 384 df-an 385 |
This theorem is referenced by: ccased 985 ccase2 986 undif3OLD 3848 ssprsseq 4297 injresinjlem 12450 prodmo 14505 nn0gcdsq 15298 symgextf1 17664 cnmsgnsubg 19742 kelac2lem 36652 |
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