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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 3398 | . 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 1289 ifcif 3393 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-11 1442 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-if 3394 |
This theorem is referenced by: eqifdc 3425 fimax2gtrilemstep 6614 updjudhcoinlf 6769 fodjuomnilem0 6800 iseqf1olemnab 9913 iseqf1olemab 9914 iseqf1olemqk 9919 iseqf1olemfvp 9922 seq3f1olemqsumkj 9923 seq3f1olemqsum 9925 seq3f1oleml 9928 seq3f1o 9929 fser0const 9947 expnnval 9954 2zsupmax 10653 isummolem3 10766 isummolem2a 10767 iisum 10771 fisum 10774 isumss 10779 fsumcl2lem 10788 fsumadd 10796 fsummulc2 10838 cvgratz 10922 ef0lem 10946 gcdval 11225 ressid2 11546 nninfalllemn 11853 nninfsellemeq 11861 nninfsellemeqinf 11863 |
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