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Mirrors > Home > MPE Home > Th. List > mt3d | Structured version Visualization version GIF version |
Description: Modus tollens deduction. (Contributed by NM, 26-Mar-1995.) |
Ref | Expression |
---|---|
mt3d.1 | ⊢ (𝜑 → ¬ 𝜒) |
mt3d.2 | ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) |
Ref | Expression |
---|---|
mt3d | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mt3d.1 | . 2 ⊢ (𝜑 → ¬ 𝜒) | |
2 | mt3d.2 | . . 3 ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) | |
3 | 2 | con1d 145 | . 2 ⊢ (𝜑 → (¬ 𝜒 → 𝜓)) |
4 | 1, 3 | mpd 15 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem is referenced by: mt3i 149 olcnd 877 ecase23d 1472 disjss3 5147 nnsuc 7905 poxp2 8167 frrlem14 8323 unxpdomlem2 9285 oismo 9578 cnfcom3lem 9741 rankelb 9862 fin33i 10407 isf34lem4 10415 canthp1lem2 10691 gchdju1 10694 pwfseqlem3 10698 inttsk 10812 r1tskina 10820 nqereu 10967 zbtwnre 12986 discr1 14275 seqcoll2 14501 bitsfzo 16469 bitsf1 16480 eucalglt 16619 4sqlem17 16995 4sqlem18 16996 ramubcl 17052 psgnunilem5 19527 odnncl 19578 gexnnod 19621 sylow1lem1 19631 torsubg 19887 prmcyg 19927 ablfacrplem 20100 pgpfac1lem2 20110 pgpfac1lem3a 20111 pgpfac1lem3 20112 xrsdsreclblem 21448 prmirredlem 21501 ppttop 23030 pptbas 23031 regr1lem 23763 alexsublem 24068 reconnlem1 24862 metnrmlem1a 24894 vitalilem4 25660 vitalilem5 25661 itg2gt0 25810 rollelem 26042 lhop1lem 26067 coefv0 26302 plyexmo 26370 lgamucov 27096 ppinprm 27210 chtnprm 27212 lgsdir 27391 lgseisenlem1 27434 2sqlem7 27483 2sqblem 27490 pntpbnd1 27645 madebdaylemlrcut 27952 dfon2lem8 35772 poimirlem25 37632 fdc 37732 ac6s6 38159 2atm 39510 llnmlplnN 39522 trlval3 40170 cdleme0moN 40208 cdleme18c 40276 qirropth 42896 aacllem 49032 |
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