ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ifiddc GIF version

Theorem ifiddc 3580
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 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)
42, 3jaoi 717 . 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 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
  Copyright terms: Public domain W3C validator