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Theorem iftrue 3479
Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
iftrue (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)

Proof of Theorem iftrue
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dedlema 953 . . 3 (𝜑 → (𝑥𝐴 ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))))
21abbi2dv 2258 . 2 (𝜑𝐴 = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))})
3 df-if 3475 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
42, 3syl6reqr 2191 1 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 697   = wceq 1331  wcel 1480  {cab 2125  ifcif 3474
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 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-if 3475
This theorem is referenced by:  iftruei  3480  iftrued  3481  ifsbdc  3486  ifcldadc  3501  ifbothdadc  3503  ifbothdc  3504  ifiddc  3505  ifcldcd  3507  ifandc  3508  fidifsnen  6764  nnnninf  7023  mkvprop  7032  uzin  9370  fzprval  9874  fztpval  9875  modifeq2int  10171  bcval  10507  bcval2  10508  sumrbdclem  11158  fsum3cvg  11159  summodclem2a  11162  isumss2  11174  fsum3ser  11178  fsumsplit  11188  sumsplitdc  11213  prodrbdclem  11352  fproddccvg  11353  flodddiv4  11642  gcd0val  11660  dfgcd2  11713  eucalgf  11747  eucalginv  11748  eucalglt  11749  unct  11966  dvexp2  12859  nnsf  13285  nninfsellemsuc  13294
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