![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mpd3an23 | Structured version Visualization version GIF version |
Description: An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
Ref | Expression |
---|---|
mpd3an23.1 | ⊢ (𝜑 → 𝜓) |
mpd3an23.2 | ⊢ (𝜑 → 𝜒) |
mpd3an23.3 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) |
Ref | Expression |
---|---|
mpd3an23 | ⊢ (𝜑 → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | mpd3an23.1 | . 2 ⊢ (𝜑 → 𝜓) | |
3 | mpd3an23.2 | . 2 ⊢ (𝜑 → 𝜒) | |
4 | mpd3an23.3 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → 𝜃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 |
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 df-3an 1088 |
This theorem is referenced by: rankcf 10815 bcpasc 14357 sqreulem 15395 qnumdencoprm 16779 qeqnumdivden 16780 xpsaddlem 17620 xpsvsca 17624 xpsle 17626 grpinvid 19030 qus0 19220 ghmid 19253 lsm01 19704 frgpadd 19796 abvneg 20844 lsmcv 21161 qusmul2idl 21307 discmp 23422 kgenhaus 23568 idnghm 24780 xmetdcn2 24873 pi1addval 25095 ipcau2 25282 gausslemma2dlem1a 27424 2lgs 27466 etasslt2 27874 uhgrsubgrself 29312 wlkl0 30396 mhmimasplusg 33026 lmhmimasvsca 33027 rlocaddval 33255 rlocmulval 33256 qusvsval 33360 carsgclctunlem2 34301 carsgclctun 34303 ballotlem1ri 34516 satefvfmla0 35403 satefvfmla1 35410 ftc1anclem5 37684 opoc1 39184 opoc0 39185 dochsat 41366 lcfrlem9 41533 fisdomnn 42264 pellfundex 42874 mnringmulrcld 44224 0ellimcdiv 45605 add2cncf 45858 stoweidlem21 45977 stoweidlem23 45979 stoweidlem32 45988 stoweidlem36 45992 stoweidlem40 45996 stoweidlem41 45997 natglobalincr 46831 mod42tp1mod8 47527 lincval0 48261 |
Copyright terms: Public domain | W3C validator |