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Mirrors > Home > MPE Home > Th. List > ad5ant125 | Structured version Visualization version GIF version |
Description: Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 23-Jun-2022.) |
Ref | Expression |
---|---|
ad5ant.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
ad5ant125 | ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ad5ant.1 | . . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
2 | 1 | 3expia 1122 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃)) |
3 | 2 | 2a1d 26 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜏 → (𝜂 → (𝜒 → 𝜃)))) |
4 | 3 | imp41 429 | 1 ⊢ (((((𝜑 ∧ 𝜓) ∧ 𝜏) ∧ 𝜂) ∧ 𝜒) → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 |
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 210 df-an 400 df-3an 1090 |
This theorem is referenced by: supxrge 42437 hoidmvlelem3 43699 |
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