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Theorem iffalsed 3369
Description: Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
iffalsed.1 (𝜑 → ¬ 𝜒)
Assertion
Ref Expression
iffalsed (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)

Proof of Theorem iffalsed
StepHypRef Expression
1 iffalsed.1 . 2 (𝜑 → ¬ 𝜒)
2 iffalse 3367 . 2 𝜒 → if(𝜒, 𝐴, 𝐵) = 𝐵)
31, 2syl 14 1 (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  ifcif 3359
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-if 3360
This theorem is referenced by:  fzprval  9175  expinnval  9576  expnegap0  9581  isumrblem  10337  gcdval  10495  eucalgf  10581  eucalginv  10582  eucalglt  10583
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