Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iffalsei Structured version   Visualization version   GIF version

Theorem iffalsei 4233
 Description: Inference associated with iffalse 4232. (Contributed by BJ, 7-Oct-2018.)
Hypothesis
Ref Expression
iffalsei.1 ¬ 𝜑
Assertion
Ref Expression
iffalsei if(𝜑, 𝐴, 𝐵) = 𝐵

Proof of Theorem iffalsei
StepHypRef Expression
1 iffalsei.1 . 2 ¬ 𝜑
2 iffalse 4232 . 2 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
31, 2ax-mp 5 1 if(𝜑, 𝐴, 𝐵) = 𝐵
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1630  ifcif 4223 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-if 4224 This theorem is referenced by:  sum0  14659  prod0  14879  prmo4  16041  prmo6  16043  itg0  23765  vieta1lem2  24285  vtxval0  26151  iedgval0  26152  ex-prmo  27652  dfrdg2  32031  dfrdg4  32389  fwddifnp1  32603  bj-pr21val  33326  bj-pr22val  33332  clsk1indlem4  38861  clsk1indlem1  38862  refsum2cnlem1  39712  limsup10ex  40517  iblempty  40692  fouriersw  40959
 Copyright terms: Public domain W3C validator