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Mirrors > Home > MPE Home > Th. List > nsyl2 | Structured version Visualization version GIF version |
Description: A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 14-Nov-2023.) |
Ref | Expression |
---|---|
nsyl2.1 | ⊢ (𝜑 → ¬ 𝜓) |
nsyl2.2 | ⊢ (¬ 𝜒 → 𝜓) |
Ref | Expression |
---|---|
nsyl2 | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsyl2.1 | . . 3 ⊢ (𝜑 → ¬ 𝜓) | |
2 | nsyl2.2 | . . 3 ⊢ (¬ 𝜒 → 𝜓) | |
3 | 1, 2 | nsyl3 138 | . 2 ⊢ (¬ 𝜒 → ¬ 𝜑) |
4 | 3 | con4i 114 | 1 ⊢ (𝜑 → 𝜒) |
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: con1i 147 oprcl 4904 epelg 5590 elfvdm 6944 ovrcl 7472 elfvov1 7473 elfvov2 7474 tfi 7874 limom 7903 oaabs2 8686 ecexr 8749 elpmi 8885 elmapex 8887 pmresg 8909 pmsspw 8916 ixpssmap2g 8966 ixpssmapg 8967 resixpfo 8975 infensuc 9194 pm54.43lem 10038 alephnbtwn 10109 cfpwsdom 10622 elbasfv 17251 elbasov 17252 restsspw 17478 homarcl 18082 isipodrs 18595 grpidval 18687 efgrelexlema 19782 subcmn 19870 dvdsrval 20378 elocv 21704 mvrf1 22024 pf1rcl 22369 matrcl 22432 restrcl 23181 ssrest 23200 iscnp2 23263 isfcls 24033 isnghm 24760 dchrrcl 27299 sltval2 27716 sltres 27722 clwwlknnn 30062 hmdmadj 31969 indispconn 35219 cvmtop1 35245 cvmtop2 35246 mrsub0 35501 mrsubf 35502 mrsubccat 35503 mrsubcn 35504 mrsubco 35506 mrsubvrs 35507 msubf 35517 mclsrcl 35546 dfon2lem7 35771 funpartlem 35924 rankeq1o 36153 bj-brrelex12ALT 37050 bj-fvimacnv0 37269 atbase 39271 llnbase 39492 lplnbase 39517 lvolbase 39561 lhpbase 39981 mapco2g 42702 |
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