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| Mirrors > Home > MPE Home > Th. List > 3jca | Structured version Visualization version GIF version | ||
| Description: Join consequents with conjunction. (Contributed by NM, 9-Apr-1994.) |
| Ref | Expression |
|---|---|
| 3jca.1 | ⊢ (𝜑 → 𝜓) |
| 3jca.2 | ⊢ (𝜑 → 𝜒) |
| 3jca.3 | ⊢ (𝜑 → 𝜃) |
| Ref | Expression |
|---|---|
| 3jca | ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3jca.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | 3jca.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
| 3 | 3jca.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
| 4 | 1, 2, 3 | jca31 514 | . 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∧ 𝜃)) |
| 5 | df-3an 1089 | . 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜃)) | |
| 6 | 4, 5 | sylibr 234 | 1 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃)) |
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