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Mirrors > Home > MPE Home > Th. List > ifid | Structured version Visualization version GIF version |
Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
Ref | Expression |
---|---|
ifid | ⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 4539 | . 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
2 | iffalse 4542 | . 2 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
3 | 1, 2 | pm2.61i 182 | 1 ⊢ if(𝜑, 𝐴, 𝐴) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ifcif 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-if 4534 |
This theorem is referenced by: csbif 4590 rabsnif 4732 somincom 6146 fsuppmptif 9442 supsn 9515 infsn 9548 wemaplem2 9590 cantnflem1 9732 xrmaxeq 13212 xrmineq 13213 xaddpnf1 13259 xaddmnf1 13261 rexmul 13304 max0add 15315 sumz 15726 prod1 15946 1arithlem4 16928 xpscf 17580 mgm2nsgrplem2 18909 mgm2nsgrplem3 18910 dmdprdsplitlem 20037 fczpsrbag 21920 mplcoe1 22044 mplcoe3 22045 mplcoe5 22047 evlslem2 22094 mdet0 22599 mdetralt2 22602 mdetunilem9 22613 madurid 22637 decpmatid 22763 cnmpopc 24940 pcoval2 25034 pcorevlem 25044 itgz 25801 itgvallem3 25806 iblposlem 25812 iblss2 25826 itgss 25832 ditg0 25873 cnplimc 25907 limcco 25913 dvexp3 26001 ply1nzb 26150 plyeq0lem 26237 dgrcolem2 26302 plydivlem4 26324 radcnv0 26445 efrlim 26997 efrlimOLD 26998 mumullem2 27208 lgsval2lem 27336 lgsdilem2 27362 fsuppind 42062 dgrsub2 42796 sqrtcval 43308 relexp1idm 43381 relexp0idm 43382 |
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