![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > impbid2 | GIF version |
Description: Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.) |
Ref | Expression |
---|---|
impbid2.1 | ⊢ (𝜓 → 𝜒) |
impbid2.2 | ⊢ (𝜑 → (𝜒 → 𝜓)) |
Ref | Expression |
---|---|
impbid2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impbid2.2 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) | |
2 | impbid2.1 | . . 3 ⊢ (𝜓 → 𝜒) | |
3 | 1, 2 | impbid1 142 | . 2 ⊢ (𝜑 → (𝜒 ↔ 𝜓)) |
4 | 3 | bicomd 141 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: biimt 241 mtt 685 biorf 744 biorfi 746 pm4.72 827 con34bdc 871 notnotbdc 872 dfandc 884 imanst 888 dfordc 892 dfor2dc 895 pm4.79dc 903 orimdidc 906 pm5.54dc 918 pm5.62dc 945 bimsc1 963 modc 2069 euan 2082 exmoeudc 2089 nebidc 2427 cgsexg 2772 cgsex2g 2773 cgsex4g 2774 elab3gf 2887 abidnf 2905 elsn2g 3625 difsn 3729 prel12 3771 dfnfc2 3827 intmin4 3872 dfiin2g 3919 elpw2g 4156 ordsucg 4501 ssrel 4714 ssrel2 4716 ssrelrel 4726 releldmb 4864 relelrnb 4865 cnveqb 5084 dmsnopg 5100 relcnvtr 5148 relcnvexb 5168 f1ocnvb 5475 ffvresb 5679 fconstfvm 5734 fnoprabg 5975 dfsmo2 6287 nntri2 6494 nntri1 6496 en1bg 6799 fieq0 6974 djulclb 7053 ismkvnex 7152 nngt1ne1 8953 znegclb 9285 iccneg 9988 fzsn 10065 fz1sbc 10095 fzofzp1b 10227 ceilqidz 10315 flqeqceilz 10317 reim0b 10870 rexanre 11228 dvdsext 11860 zob 11895 pc11 12329 pcz 12330 issubg2m 13047 issubg4m 13051 opprringbg 13248 issubrg2 13360 eltg3 13527 bastop 13545 cnptoprest 13709 cos11 14244 zabsle1 14370 lgsabs1 14410 bj-om 14659 |
Copyright terms: Public domain | W3C validator |