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Mirrors > Home > MPE Home > Th. List > 3simpb | Structured version Visualization version GIF version |
Description: Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) |
Ref | Expression |
---|---|
3simpb | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3ancomb 1040 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓)) | |
2 | 3simpa 1051 | . 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜓) → (𝜑 ∧ 𝜒)) | |
3 | 1, 2 | sylbi 206 | 1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜑 ∧ 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: 3adant2 1073 3adantl2 1211 3adantr2 1214 cfcof 8979 axcclem 9162 enqeq 9635 ltleletr 10009 ixxssixx 12060 prodmolem2 14504 prodmo 14505 zprod 14506 muldvds1 14844 dvds2add 14853 dvds2sub 14854 dvdstr 14856 initoeu2lem2 16488 pospropd 16957 csrgbinom 18369 smadiadetglem2 20297 ismbf3d 23227 mbfi1flimlem 23295 colinearalg 25590 2pthon 26132 constr3trllem2 26179 wlkiswwlkinj 26246 2wlkonotv 26404 usg2spthonot 26415 usg2spthonot0 26416 vdn1frgrav2 26552 vdgn1frgrav2 26553 bnj967 30269 bnj1110 30304 cgr3permute3 31324 cgr3com 31330 brofs2 31354 areacirclem4 32673 paddasslem14 34137 lhpexle1 34312 cdlemk19w 35278 ismrc 36282 iocinico 36816 gneispb 37449 fourierdlem113 39112 sigaras 39693 sigarms 39694 leltletr 39940 fpropnf1 40337 frusgrnn0 40771 wlkwwlkinj 41093 21wlkond 41144 2pthond 41149 2pthon3v-av 41150 umgr2adedgwlkonALT 41154 usgr2wspthons3 41167 vdgn1frgrv2 41466 av-extwwlkfab 41520 lincresunit3lem3 42057 lincresunit3 42064 |
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