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Mirrors > Home > ILE Home > Th. List > ifiddc | GIF version |
Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.) |
Ref | Expression |
---|---|
ifiddc | ⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmiddc 831 | . 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | iftrue 3531 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
3 | iffalse 3534 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
4 | 2, 3 | jaoi 711 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → if(𝜑, 𝐴, 𝐴) = 𝐴) |
5 | 1, 4 | syl 14 | 1 ⊢ (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 703 DECID wdc 829 = wceq 1348 ifcif 3526 |
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 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-if 3527 |
This theorem is referenced by: xaddpnf1 9803 xaddmnf1 9805 isumz 11352 prod1dc 11549 1arithlem4 12318 lgsval2lem 13705 lgsdilem2 13731 |
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