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Mirrors > Home > MPE Home > Th. List > jaodan | Structured version Visualization version GIF version |
Description: Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
jaodan.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
jaodan.2 | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Ref | Expression |
---|---|
jaodan | ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaodan.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | 1 | ex 416 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
3 | jaodan.2 | . . . 4 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) | |
4 | 3 | ex 416 | . . 3 ⊢ (𝜑 → (𝜃 → 𝜒)) |
5 | 2, 4 | jaod 859 | . 2 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → 𝜒)) |
6 | 5 | imp 410 | 1 ⊢ ((𝜑 ∧ (𝜓 ∨ 𝜃)) → 𝜒) |
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