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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 414 | . 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 397 |
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 206 df-an 398 |
This theorem is referenced by: dfsb2 2493 xpcan 6176 xpcan2 6177 sucdom2OLD 9082 mapxpen 9143 sucdom2 9206 f1finf1oOLD 9272 unfiOLD 9313 inf3lem3 9625 dfac12r 10141 nnadju 10192 cfsuc 10252 fin23lem26 10320 iundom2g 10535 inar1 10770 rankcf 10772 ltsrpr 11072 supsrlem 11106 axpre-sup 11164 nominpos 12449 ublbneg 12917 qbtwnre 13178 fsequb 13940 fi1uzind 14458 brfi1indALT 14461 ccats1pfxeqrex 14665 rexanre 15293 rexuzre 15299 rexico 15300 caubnd 15305 rlim2lt 15441 rlim3 15442 lo1bddrp 15469 o1lo1 15481 climshftlem 15518 rlimcn3 15534 rlimo1 15561 lo1add 15571 lo1mul 15572 lo1le 15598 isercoll 15614 serf0 15627 cvgcmp 15762 dvds1lem 16211 dvds2lem 16212 mulmoddvds 16273 isprm5 16644 vdwlem2 16915 vdwlem10 16923 vdwlem11 16924 lsmcv 20754 lmconst 22765 ptcnplem 23125 fclscmp 23534 tsmsres 23648 addcnlem 24380 lebnumlem3 24479 xlebnum 24481 lebnumii 24482 iscmet3lem2 24809 bcthlem4 24844 cniccbdd 24978 ovoliunlem2 25020 mbfi1flimlem 25240 ply1divex 25654 aalioulem3 25847 aalioulem5 25849 aalioulem6 25850 aaliou 25851 ulmshftlem 25901 ulmbdd 25910 tanarg 26127 cxploglim 26482 ftalem2 26578 ftalem7 26583 dchrisumlem3 26994 frgrogt3nreg 29650 ubthlem3 30125 spansncol 30821 riesz1 31318 fineqvac 34097 erdsze2lem2 34195 dfrdg4 34923 neibastop2 35246 onsuct0 35326 bj-bary1 36193 topdifinffinlem 36228 finorwe 36263 poimirlem24 36512 incsequz 36616 caushft 36629 equivbnd 36658 cntotbnd 36664 4atexlemex4 38944 frege124d 42512 gneispace 42885 expgrowth 43094 vk15.4j 43289 sstrALT2 43596 iccpartdisj 46105 fppr2odd 46399 |
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