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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 837 | . 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑)) | |
2 | iftrue 3551 | . . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
3 | iffalse 3554 | . . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴) | |
4 | 2, 3 | jaoi 717 | . 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 709 DECID wdc 835 = wceq 1363 ifcif 3546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-11 1516 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-if 3547 |
This theorem is referenced by: xaddpnf1 9860 xaddmnf1 9862 isumz 11411 prod1dc 11608 1arithlem4 12378 xpscf 12785 lgsval2lem 14764 lgsdilem2 14790 |
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