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Mirrors > Home > MPE Home > Th. List > iftruei | Structured version Visualization version GIF version |
Description: Inference associated with iftrue 4537. (Contributed by BJ, 7-Oct-2018.) |
Ref | Expression |
---|---|
iftruei.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
iftruei | ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftruei.1 | . 2 ⊢ 𝜑 | |
2 | iftrue 4537 | . 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ if(𝜑, 𝐴, 𝐵) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ifcif 4531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-if 4532 |
This theorem is referenced by: oe0m 8555 ttrcltr 9754 ttukeylem4 10550 xnegpnf 13248 xnegmnf 13249 xaddpnf1 13265 xaddpnf2 13266 xaddmnf1 13267 xaddmnf2 13268 pnfaddmnf 13269 mnfaddpnf 13270 xmul01 13306 exp0 14103 swrd00 14679 sgn0 15125 lcm0val 16628 prmo2 17074 prmo3 17075 prmo5 17163 mulg0 19105 zzngim 21589 obsipid 21760 mvrid 22022 mamulid 22463 mamurid 22464 mat1dimid 22496 scmatf1 22553 mdetdiagid 22622 chpdmatlem3 22862 chpidmat 22869 fclscmpi 24053 ioorinv 25625 ig1pval2 26231 dgrcolem2 26329 plydivlem4 26353 vieta1lem2 26368 0cxp 26723 cxpexp 26725 lgs0 27369 lgs2 27373 2lgs2 27464 left1s 27948 exps0 28425 axlowdim 28991 1loopgrvd2 29536 eupth2 30268 ex-prmo 30488 madjusmdetlem1 33788 signsw0glem 34547 breprexp 34627 ex-sategoelel 35406 rdgprc0 35775 bj-pr11val 36988 bj-pr22val 37002 mapdhval0 41708 hdmap1val0 41782 refsum2cnlem1 44975 liminf10ex 45730 cncfiooicclem1 45849 fouriersw 46187 hspmbllem1 46582 blen0 48422 0dig1 48459 |
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