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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 824 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜓) → 𝜏) |
4 | ccase.3 | . . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏) | |
5 | ccase.4 | . . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏) | |
6 | 4, 5 | jaoian 824 | . 2 ⊢ (((𝜑 ∨ 𝜒) ∧ 𝜃) → 𝜏) |
7 | 3, 6 | jaodan 826 | 1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 |
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 197 df-or 385 df-an 386 |
This theorem is referenced by: ccased 988 ccase2 989 undif3OLD 3889 ssprsseq 4357 injresinjlem 12588 prodmo 14666 nn0gcdsq 15460 symgextf1 17841 cnmsgnsubg 19923 kelac2lem 37634 |
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