Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1257 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜓) → 𝜒) |
3 | 2 | anabss3 860 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: supsn 8261 infsn 8293 grusn 9505 subeq0 10186 halfaddsub 11142 avglt2 11148 modabs2 12566 efsub 14669 sinmul 14741 divalgmod 14967 divalgmodOLD 14968 modgcd 15091 pythagtriplem4 15362 pythagtriplem16 15373 pltirr 16786 latjidm 16897 latmidm 16909 ipopos 16983 mulgmodid 17404 f1omvdcnv 17687 lsmss1 17902 zntoslem 19724 obsipid 19885 smadiadetlem2 20289 smadiadet 20295 ordtt1 20993 xmet0 21957 nmsq 22802 tchcphlem3 22840 tchcph 22844 grpoidinvlem1 26742 grpodivid 26780 nvmid 26898 ipidsq 26949 5oalem1 27897 3oalem2 27906 unopf1o 28159 unopnorm 28160 hmopre 28166 ballotlemfc0 29881 ballotlemfcc 29882 pdivsq 30888 gcdabsorb 30891 fv2ndcnv 30926 cgr3rflx 31331 endofsegid 31362 tailini 31541 nnssi2 31624 nndivlub 31627 opoccl 33499 opococ 33500 opexmid 33512 opnoncon 33513 cmtidN 33562 ltrniotaidvalN 34889 pell14qrexpclnn0 36448 rmxdbl 36522 rmydbl 36523 rhmsubclem3 41880 rhmsubcALTVlem3 41899 |
Copyright terms: Public domain | W3C validator |