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Mirrors > Home > ILE Home > Th. List > 3jaodan | GIF version |
Description: Disjunction of 3 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 114 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | 3jaodan.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
4 | 3 | ex 114 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) |
5 | 3jaodan.3 | . . . 4 ⊢ ((𝜑 ∧ 𝜏) → 𝜒) | |
6 | 5 | ex 114 | . . 3 ⊢ (𝜑 → (𝜏 → 𝜒)) |
7 | 2, 4, 6 | 3jaod 1282 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜏) → 𝜒)) |
8 | 7 | imp 123 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃 ∨ 𝜏)) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ w3o 961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 |
This theorem is referenced by: zeo 9156 xrltnsym 9579 xrlttr 9581 xrltso 9582 xrlttri3 9583 xltnegi 9618 xaddcom 9644 xnegdi 9651 xsubge0 9664 qbtwnxr 10035 blssioo 12714 |
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