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Mirrors > Home > NFE Home > Th. List > anidm | GIF version |
Description: Idempotent law for conjunction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Mar-2014.) |
Ref | Expression |
---|---|
anidm | ⊢ ((φ ∧ φ) ↔ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm4.24 624 | . 2 ⊢ (φ ↔ (φ ∧ φ)) | |
2 | 1 | bicomi 193 | 1 ⊢ ((φ ∧ φ) ↔ φ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 |
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 177 df-an 360 |
This theorem is referenced by: anidmdbi 627 anandi 801 anandir 802 nannot 1293 truantru 1336 falanfal 1339 nic-axALT 1439 sbnf2 2108 2eu4 2287 elcomplg 3218 inidm 3464 ncfinlower 4483 nnpw1ex 4484 nnpweq 4523 phialllem1 4616 xp11 5056 fununi 5160 |
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