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Mirrors > Home > MPE Home > Th. List > 3jaoian | Structured version Visualization version GIF version |
Description: Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
3jaoian.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3jaoian.2 | ⊢ ((𝜃 ∧ 𝜓) → 𝜒) |
3jaoian.3 | ⊢ ((𝜏 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
3jaoian | ⊢ (((𝜑 ∨ 𝜃 ∨ 𝜏) ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3jaoian.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 449 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 3jaoian.2 | . . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒) | |
4 | 3 | ex 449 | . . 3 ⊢ (𝜃 → (𝜓 → 𝜒)) |
5 | 3jaoian.3 | . . . 4 ⊢ ((𝜏 ∧ 𝜓) → 𝜒) | |
6 | 5 | ex 449 | . . 3 ⊢ (𝜏 → (𝜓 → 𝜒)) |
7 | 2, 4, 6 | 3jaoi 1383 | . 2 ⊢ ((𝜑 ∨ 𝜃 ∨ 𝜏) → (𝜓 → 𝜒)) |
8 | 7 | imp 444 | 1 ⊢ (((𝜑 ∨ 𝜃 ∨ 𝜏) ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∨ w3o 1030 |
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 df-3or 1032 df-3an 1033 |
This theorem is referenced by: xrltnsym 11846 xrlttri 11848 xrlttr 11849 qbtwnxr 11905 xltnegi 11921 xaddcom 11945 xnegdi 11950 lcmftp 15187 xaddeq0 28907 3ccased 30855 |
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