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Mirrors > Home > ILE Home > Th. List > iftrued | GIF version |
Description: Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iftrued.1 | ⊢ (𝜑 → 𝜒) |
Ref | Expression |
---|---|
iftrued | ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrued.1 | . 2 ⊢ (𝜑 → 𝜒) | |
2 | iftrue 3443 | . 2 ⊢ (𝜒 → if(𝜒, 𝐴, 𝐵) = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ifcif 3438 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-if 3439 |
This theorem is referenced by: eqifdc 3470 mposnif 5817 fimax2gtrilemstep 6745 updjudhcoinlf 6915 omp1eomlem 6929 difinfsnlem 6934 ctssdclemn0 6945 ctssdc 6948 enumctlemm 6949 fodju0 6967 iseqf1olemnab 10148 iseqf1olemab 10149 iseqf1olemqk 10154 iseqf1olemfvp 10157 seq3f1olemqsumkj 10158 seq3f1olemqsum 10160 seq3f1oleml 10163 seq3f1o 10164 fser0const 10176 expnnval 10183 2zsupmax 10883 xrmaxifle 10901 xrmaxiflemab 10902 xrmaxiflemlub 10903 xrmaxiflemcom 10904 summodclem3 11035 summodclem2a 11036 isum 11040 fsum3 11042 isumss 11046 fsumcl2lem 11053 fsumadd 11061 fsummulc2 11103 cvgratz 11187 ef0lem 11211 gcdval 11490 ennnfonelemss 11762 ennnfonelemkh 11764 ennnfonelemhf1o 11765 ressid2 11855 nninfalllemn 12883 nninfsellemeq 12891 nninfsellemeqinf 12893 nninffeq 12897 |
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