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Mirrors > Home > ILE Home > Th. List > 3jaoian | GIF version |
Description: Disjunction of 3 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 115 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 3jaoian.2 | . . . 4 ⊢ ((𝜃 ∧ 𝜓) → 𝜒) | |
4 | 3 | ex 115 | . . 3 ⊢ (𝜃 → (𝜓 → 𝜒)) |
5 | 3jaoian.3 | . . . 4 ⊢ ((𝜏 ∧ 𝜓) → 𝜒) | |
6 | 5 | ex 115 | . . 3 ⊢ (𝜏 → (𝜓 → 𝜒)) |
7 | 2, 4, 6 | 3jaoi 1303 | . 2 ⊢ ((𝜑 ∨ 𝜃 ∨ 𝜏) → (𝜓 → 𝜒)) |
8 | 7 | imp 124 | 1 ⊢ (((𝜑 ∨ 𝜃 ∨ 𝜏) ∧ 𝜓) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ w3o 977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 |
This theorem is referenced by: xrltnsym 9791 xrlttr 9793 xltnegi 9833 xaddcom 9859 xnegdi 9866 qbtwnxr 10255 |
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