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

Theorem ifiddc 3559
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 831 . 2 (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2 iftrue 3531 . . 3 (𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
3 iffalse 3534 . . 3 𝜑 → if(𝜑, 𝐴, 𝐴) = 𝐴)
42, 3jaoi 711 . 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 703  DECID wdc 829   = wceq 1348  ifcif 3526
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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-dc 830  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-if 3527
This theorem is referenced by:  xaddpnf1  9803  xaddmnf1  9805  isumz  11352  prod1dc  11549  1arithlem4  12318  lgsval2lem  13705  lgsdilem2  13731
  Copyright terms: Public domain W3C validator