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Mirrors > Home > MPE Home > Th. List > ifeq1d | Structured version Visualization version GIF version |
Description: Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.) |
Ref | Expression |
---|---|
ifeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ifeq1d | ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ifeq1 4040 | . 2 ⊢ (𝐴 = 𝐵 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ifcif 4036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-un 3545 df-if 4037 |
This theorem is referenced by: ifeq12d 4056 ifbieq1d 4059 ifeq1da 4066 rabsnif 4202 fsuppmptif 8188 cantnflem1 8469 sumeq2w 14270 cbvsum 14273 isumless 14416 prodss 14516 subgmulg 17431 evlslem2 19333 dmatcrng 20127 scmatscmiddistr 20133 scmatcrng 20146 marrepfval 20185 mdetr0 20230 mdetunilem8 20244 madufval 20262 madugsum 20268 minmar1fval 20271 decpmatid 20394 monmatcollpw 20403 pmatcollpwscmatlem1 20413 cnmpt2pc 22535 pcoval2 22624 pcopt 22630 itgz 23353 iblss2 23378 itgss 23384 itgcn 23415 plyeq0lem 23770 dgrcolem2 23834 plydivlem4 23855 leibpi 24469 chtublem 24736 sumdchr 24797 bposlem6 24814 lgsval 24826 dchrvmasumiflem2 24991 padicabvcxp 25121 dfrdg3 30946 matunitlindflem1 32575 ftc1anclem2 32656 ftc1anclem5 32659 ftc1anclem7 32661 hoidifhspval 39498 hoimbl 39521 |
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