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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 2773 cgsex2g 2774 cgsex4g 2775 elab3gf 2888 abidnf 2906 elsn2g 3626 difsn 3730 prel12 3772 dfnfc2 3828 intmin4 3873 dfiin2g 3920 elpw2g 4157 ordsucg 4502 ssrel 4715 ssrel2 4717 ssrelrel 4727 releldmb 4865 relelrnb 4866 cnveqb 5085 dmsnopg 5101 relcnvtr 5149 relcnvexb 5169 f1ocnvb 5476 ffvresb 5680 fconstfvm 5735 fnoprabg 5976 dfsmo2 6288 nntri2 6495 nntri1 6497 en1bg 6800 fieq0 6975 djulclb 7054 ismkvnex 7153 nngt1ne1 8954 znegclb 9286 iccneg 9989 fzsn 10066 fz1sbc 10096 fzofzp1b 10228 ceilqidz 10316 flqeqceilz 10318 reim0b 10871 rexanre 11229 dvdsext 11861 zob 11896 pc11 12330 pcz 12331 issubg2m 13049 issubg4m 13053 opprringbg 13250 issubrg2 13362 eltg3 13560 bastop 13578 cnptoprest 13742 cos11 14277 zabsle1 14403 lgsabs1 14443 bj-om 14692 |
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