![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > syl6an | Structured version Visualization version GIF version |
Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Ref | Expression |
---|---|
syl6an.1 | ⊢ (𝜑 → 𝜓) |
syl6an.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
syl6an.3 | ⊢ ((𝜓 ∧ 𝜃) → 𝜏) |
Ref | Expression |
---|---|
syl6an | ⊢ (𝜑 → (𝜒 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6an.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl6an.2 | . 2 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
3 | syl6an.3 | . . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏) | |
4 | 3 | ex 412 | . 2 ⊢ (𝜓 → (𝜃 → 𝜏)) |
5 | 1, 2, 4 | sylsyld 61 | 1 ⊢ (𝜑 → (𝜒 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 |
This theorem is referenced by: dfsb2 2495 xpcan 6197 xpcan2 6198 sucdom2OLD 9120 mapxpen 9181 sucdom2 9240 f1finf1oOLD 9303 inf3lem3 9667 dfac12r 10184 nnadju 10235 cfsuc 10294 fin23lem26 10362 iundom2g 10577 inar1 10812 rankcf 10814 ltsrpr 11114 supsrlem 11148 axpre-sup 11206 nominpos 12500 ublbneg 12972 qbtwnre 13237 fsequb 14012 fi1uzind 14542 brfi1indALT 14545 ccats1pfxeqrex 14749 rexanre 15381 rexuzre 15387 rexico 15388 caubnd 15393 rlim2lt 15529 rlim3 15530 lo1bddrp 15557 o1lo1 15569 climshftlem 15606 rlimcn3 15622 rlimo1 15649 lo1add 15659 lo1mul 15660 lo1le 15684 isercoll 15700 serf0 15713 cvgcmp 15848 dvds1lem 16301 dvds2lem 16302 mulmoddvds 16363 isprm5 16740 vdwlem2 17015 vdwlem10 17023 vdwlem11 17024 lsmcv 21160 lmconst 23284 ptcnplem 23644 fclscmp 24053 tsmsres 24167 addcnlem 24899 lebnumlem3 25008 xlebnum 25010 lebnumii 25011 iscmet3lem2 25339 bcthlem4 25374 cniccbdd 25509 ovoliunlem2 25551 mbfi1flimlem 25771 ply1divex 26190 aalioulem3 26390 aalioulem5 26392 aalioulem6 26393 aaliou 26394 ulmshftlem 26446 ulmbdd 26455 tanarg 26675 cxploglim 27035 ftalem2 27131 ftalem7 27136 dchrisumlem3 27549 frgrogt3nreg 30425 ubthlem3 30900 spansncol 31596 riesz1 32093 fineqvac 35089 erdsze2lem2 35188 dfrdg4 35932 neibastop2 36343 onsuct0 36423 weiunpo 36447 bj-bary1 37294 topdifinffinlem 37329 finorwe 37364 poimirlem24 37630 incsequz 37734 caushft 37747 equivbnd 37776 cntotbnd 37782 4atexlemex4 40055 frege124d 43750 gneispace 44123 expgrowth 44330 vk15.4j 44525 sstrALT2 44832 iccpartdisj 47361 fppr2odd 47655 |
Copyright terms: Public domain | W3C validator |