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| Mirrors > Home > MPE Home > Th. List > 3anim123i | Structured version Visualization version GIF version | ||
| Description: Join antecedents and consequents with conjunction. (Contributed by NM, 8-Apr-1994.) |
| Ref | Expression |
|---|---|
| 3anim123i.1 | ⊢ (𝜑 → 𝜓) |
| 3anim123i.2 | ⊢ (𝜒 → 𝜃) |
| 3anim123i.3 | ⊢ (𝜏 → 𝜂) |
| Ref | Expression |
|---|---|
| 3anim123i | ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3anim123i.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | 1 | 3ad2ant1 1134 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜓) |
| 3 | 3anim123i.2 | . . 3 ⊢ (𝜒 → 𝜃) | |
| 4 | 3 | 3ad2ant2 1135 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜃) |
| 5 | 3anim123i.3 | . . 3 ⊢ (𝜏 → 𝜂) | |
| 6 | 5 | 3ad2ant3 1136 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂) |
| 7 | 2, 4, 6 | 3jca 1129 | 1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜃 ∧ 𝜂)) |
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