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Mirrors > Home > MPE Home > Th. List > 3jaodan | Structured version Visualization version GIF version |
Description: Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
3jaodan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3jaodan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
3jaodan.3 | ⊢ ((𝜑 ∧ 𝜏) → 𝜒) |
Ref | Expression |
---|---|
3jaodan | ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3jaodan.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 3jaodan.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
4 | 3 | ex 449 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) |
5 | 3jaodan.3 | . . . 4 ⊢ ((𝜑 ∧ 𝜏) → 𝜒) | |
6 | 5 | ex 449 | . . 3 ⊢ (𝜑 → (𝜏 → 𝜒)) |
7 | 2, 4, 6 | 3jaod 1432 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜏) → 𝜒)) |
8 | 7 | imp 444 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∨ w3o 1053 |
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 384 df-an 385 df-3or 1055 df-3an 1056 |
This theorem is referenced by: onzsl 7088 zeo 11501 xrltnsym 12008 xrlttri 12010 xrlttr 12011 qbtwnxr 12069 xltnegi 12085 xaddcom 12109 xnegdi 12116 xsubge0 12129 xrub 12180 bpoly3 14833 blssioo 22645 ismbf2d 23453 itg2seq 23554 eliccioo 29767 3ccased 31726 lineelsb2 32380 dfxlim2v 40391 |
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