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Mirrors > Home > MPE Home > Th. List > ad5ant235 | Structured version Visualization version GIF version |
Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) |
Ref | Expression |
---|---|
ad5ant235.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
ad5ant235 | ⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ad5ant235.1 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 1 | 3exp 1256 | . . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) |
3 | 2 | a1ddd 78 | . . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜃)))) |
4 | 3 | a1ddd 78 | . . . . 5 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜂 → (𝜏 → 𝜃))))) |
5 | 4 | com5r 102 | . . . 4 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → (𝜂 → 𝜃))))) |
6 | 5 | com45 95 | . . 3 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → (𝜒 → 𝜃))))) |
7 | 6 | imp 444 | . 2 ⊢ ((𝜏 ∧ 𝜑) → (𝜓 → (𝜂 → (𝜒 → 𝜃)))) |
8 | 7 | imp41 617 | 1 ⊢ (((((𝜏 ∧ 𝜑) ∧ 𝜓) ∧ 𝜂) ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-an 385 df-3an 1033 |
This theorem is referenced by: (None) |
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