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Mirrors > Home > MPE Home > Th. List > pm3.2an3 | Structured version Visualization version GIF version |
Description: Version of pm3.2 463 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Kyle Wyonch, 24-Apr-2021.) |
Ref | Expression |
---|---|
pm3.2an3 | ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1039 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
2 | 1 | biimpri 218 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → (𝜑 ∧ 𝜓 ∧ 𝜒)) |
3 | 2 | exp31 630 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 ∧ 𝜓 ∧ 𝜒)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 |
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 197 df-an 386 df-3an 1039 |
This theorem is referenced by: 3exp 1264 tratrb 38746 19.21a3con13vVD 39087 tratrbVD 39097 |
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