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Theorem ifiddc 3444
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 785 . 2 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 iftrue 3418 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3 iffalse 3421 . . 3 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
42, 3jaoi 674 . 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 667  DECID wdc 783   = wceq 1296  ifcif 3413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-dc 784  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-if 3414
This theorem is referenced by:  xaddpnf1  9412  xaddmnf1  9414  isumz  10932
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