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Mirrors > Home > MPE Home > Th. List > 3anidm23 | Structured version Visualization version GIF version |
Description: Inference from idempotent law for conjunction. (Contributed by NM, 1-Feb-2007.) |
Ref | Expression |
---|---|
3anidm23.1 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
3anidm23 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anidm23.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜓) → 𝜒) | |
2 | 1 | 3expa 1117 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒) |
3 | 2 | anabss3 675 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 |
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 207 df-an 396 df-3an 1088 |
This theorem is referenced by: supsn 9510 infsn 9543 grusn 10842 subeq0 11533 halfaddsub 12497 avglt2 12503 modabs2 13942 efsub 16133 sinmul 16205 divalgmod 16440 modgcd 16566 pythagtriplem4 16853 pythagtriplem16 16864 pltirr 18393 latjidm 18520 latmidm 18532 ipopos 18594 mulgmodid 19144 f1omvdcnv 19477 lsmss1 19698 rhmsubclem3 20704 zntoslem 21593 obsipid 21760 smadiadetlem2 22686 smadiadet 22692 ordtt1 23403 xmet0 24368 nmsq 25242 tcphcphlem3 25281 tcphcph 25285 grpoidinvlem1 30533 grpodivid 30571 nvmid 30688 ipidsq 30739 5oalem1 31683 3oalem2 31692 unopf1o 31945 unopnorm 31946 hmopre 31952 ballotlemfc0 34474 ballotlemfcc 34475 gcdabsorb 35730 cgr3rflx 36036 endofsegid 36067 tailini 36359 nnssi2 36438 nndivlub 36441 brin2 38391 opoccl 39176 opococ 39177 opexmid 39189 opnoncon 39190 cmtidN 39239 ltrniotaidvalN 40566 pell14qrexpclnn0 42854 rmxdbl 42928 rmydbl 42929 rhmsubcALTVlem3 48127 |
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