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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 822 | . 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | iftrue 3510 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
3 | iffalse 3513 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
4 | 2, 3 | jaoi 706 | . 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 698 DECID wdc 820 = wceq 1335 ifcif 3505 |
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 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-if 3506 |
This theorem is referenced by: xaddpnf1 9750 xaddmnf1 9752 isumz 11286 prod1dc 11483 |
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