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Theorem iffalsed 3341
 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 3339 . 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 1243  ifcif 3331 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-if 3332 This theorem is referenced by:  fzprval  8944  expinnval  9258  expnegap0  9263
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