Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > simprbda | Structured version Visualization version GIF version |
Description: Deduction eliminating a conjunct. (Contributed by NM, 22-Oct-2007.) |
Ref | Expression |
---|---|
pm3.26bda.1 | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Ref | Expression |
---|---|
simprbda | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.26bda.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | |
2 | 1 | biimpa 500 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) |
3 | 2 | simpld 474 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 |
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 |
This theorem is referenced by: elrabi 3328 oteqex 4889 fsnex 6438 fisupg 8093 fiinfg 8288 cantnff 8454 fseqenlem2 8731 fpwwe2lem11 9341 fpwwe2lem12 9342 fpwwe2 9344 rlimsqzlem 14227 ramub1lem2 15569 mriss 16118 invfun 16247 pltle 16784 subgslw 17854 frgpnabllem2 18100 cyggeninv 18108 ablfaclem3 18309 lmodfopnelem1 18722 psrbagf 19186 mplind 19323 pjff 19875 pjf2 19877 pjfo 19878 pjcss 19879 fvmptnn04ifc 20476 chfacfisf 20478 chfacfisfcpmat 20479 tg1 20579 cldss 20643 cnf2 20863 cncnp 20894 lly1stc 21109 refbas 21123 qtoptop2 21312 qtoprest 21330 elfm3 21564 flfelbas 21608 cnextf 21680 restutopopn 21852 cfilufbas 21903 fmucnd 21906 blgt0 22014 xblss2ps 22016 xblss2 22017 tngngp 22268 cfilfil 22873 iscau2 22883 caufpm 22888 cmetcaulem 22894 dvcnp2 23489 dvfsumrlim 23598 dvfsumrlim2 23599 fta1g 23731 dvdsflsumcom 24714 fsumvma 24738 vmadivsumb 24972 dchrisumlema 24977 dchrvmasumlem1 24984 dchrvmasum2lem 24985 dchrvmasumiflem1 24990 selbergb 25038 selberg2b 25041 pntibndlem3 25081 pntlem3 25098 motgrp 25238 oppnid 25438 clwlkiswlk 26285 sspnv 26965 lnof 26994 bloln 27023 reff 29234 signsply0 29954 cvmliftmolem1 30517 cvmlift2lem9a 30539 mbfresfi 32626 itg2gt0cn 32635 ismtyres 32777 ghomf 32859 rngoisohom 32949 pridlidl 33004 pridlnr 33005 maxidlidl 33010 lflf 33368 lkrcl 33397 cvrlt 33575 cvrle 33583 atbase 33594 llnbase 33813 lplnbase 33838 lvolbase 33882 psubssat 34058 lhpbase 34302 laut1o 34389 ldillaut 34415 ltrnldil 34426 diadmclN 35344 pell1234qrre 36434 lnmlsslnm 36669 cvgdvgrat 37534 stoweidlem34 38927 |
Copyright terms: Public domain | W3C validator |