![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > anandi3r | Structured version Visualization version GIF version |
Description: Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.) |
Ref | Expression |
---|---|
anandi3r | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anan32 1094 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜒) ∧ 𝜓)) | |
2 | anandir 675 | . 2 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓))) | |
3 | 1, 2 | bitri 274 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∧ w3a 1084 |
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 206 df-an 395 df-3an 1086 |
This theorem is referenced by: refsymrel2 38071 refsymrel3 38072 dfeqvrel2 38094 dfeqvrel3 38095 i0oii 48016 alsi-no-surprise 48307 |
Copyright terms: Public domain | W3C validator |