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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 3449 | . 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 1316 ifcif 3444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-if 3445 |
This theorem is referenced by: eqifdc 3476 mposnif 5833 fimax2gtrilemstep 6762 updjudhcoinlf 6933 omp1eomlem 6947 difinfsnlem 6952 ctssdclemn0 6963 ctssdc 6966 enumctlemm 6967 fodju0 6987 iseqf1olemnab 10229 iseqf1olemab 10230 iseqf1olemqk 10235 iseqf1olemfvp 10238 seq3f1olemqsumkj 10239 seq3f1olemqsum 10241 seq3f1oleml 10244 seq3f1o 10245 fser0const 10257 expnnval 10264 2zsupmax 10965 xrmaxifle 10983 xrmaxiflemab 10984 xrmaxiflemlub 10985 xrmaxiflemcom 10986 summodclem3 11117 summodclem2a 11118 isum 11122 fsum3 11124 isumss 11128 fsumcl2lem 11135 fsumadd 11143 fsummulc2 11185 cvgratz 11269 ef0lem 11293 gcdval 11575 ennnfonelemss 11850 ennnfonelemkh 11852 ennnfonelemhf1o 11853 ressid2 11945 subctctexmid 13123 nninfalllemn 13129 nninfsellemeq 13137 nninfsellemeqinf 13139 nninffeq 13143 |
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