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Theorem iftrue 3384
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 913 . . 3 (𝜑 → (𝑥𝐴 ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))))
21abbi2dv 2203 . 2 (𝜑𝐴 = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))})
3 df-if 3380 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
42, 3syl6reqr 2136 1 (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662   = wceq 1287  wcel 1436  {cab 2071  ifcif 3379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-if 3380
This theorem is referenced by:  iftruei  3385  iftrued  3386  ifsbdc  3391  ifcldadc  3406  ifbothdadc  3408  ifbothdc  3409  ifiddc  3410  ifcldcd  3412  ifandc  3413  fidifsnen  6538  nnnninf  6750  uzin  8983  fzprval  9426  fztpval  9427  modifeq2int  9721  expival  9855  bcval  10053  bcval2  10054  isumrblem  10655  fisumcvg  10656  isummolem2a  10660  flodddiv4  10809  gcd0val  10827  dfgcd2  10878  eucalgf  10912  eucalginv  10913  eucalglt  10914  nnsf  11333  nninfsellemsuc  11342
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