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Theorem ifiddc 3475
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 806 . 2 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 iftrue 3449 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3 iffalse 3452 . . 3 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
42, 3jaoi 690 . 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 682  DECID wdc 804   = wceq 1316  ifcif 3444
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 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-dc 805  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-if 3445
This theorem is referenced by:  xaddpnf1  9597  xaddmnf1  9599  isumz  11126
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