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Theorem iffalse 3452
Description: Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
Assertion
Ref Expression
iffalse 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Proof of Theorem iffalse
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dedlemb 939 . . 3 𝜑 → (𝑥𝐵 ↔ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))))
21abbi2dv 2236 . 2 𝜑𝐵 = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))})
3 df-if 3445 . 2 if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
42, 3syl6reqr 2169 1 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 682   = wceq 1316  wcel 1465  {cab 2103  ifcif 3444
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 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-if 3445
This theorem is referenced by:  iffalsei  3453  iffalsed  3454  ifnefalse  3455  ifsbdc  3456  ifcldadc  3471  ifeq1dadc  3472  ifbothdadc  3473  ifbothdc  3474  ifiddc  3475  ifcldcd  3477  ifandc  3478  fidifsnen  6732  nnnninf  6991  uzin  9326  modifeq2int  10127  bcval  10463  bcval3  10465  sumrbdclem  11113  fsum3cvg  11114  summodclem2a  11118  sumsplitdc  11169  flodddiv4  11558  gcdn0val  11577  dfgcd2  11629  lcmn0val  11674  unct  11881
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