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Theorem ifiddc 3538
Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
Assertion
Ref Expression
ifiddc (DECID 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)

Proof of Theorem ifiddc
StepHypRef 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)
42, 3jaoi 706 . 2 ((𝜑 ∨ ¬ 𝜑) → if(𝜑, 𝐴, 𝐴) = 𝐴)
51, 4syl 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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