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

Theorem eqif 3919
Description: Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
eqif  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )

Proof of Theorem eqif
StepHypRef Expression
1 eqeq2 2462 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  B ) )
2 eqeq2 2462 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  =  if ( ph ,  B ,  C )  <->  A  =  C ) )
31, 2elimif 3915 1  |-  ( A  =  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444   ifcif 3881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-if 3882
This theorem is referenced by:  ifval  3920  xpima  5279  fin23lem19  8766  fin23lem28  8770  fin23lem29  8771  fin23lem30  8772  aalioulem3  23290  ifbieq12d2  28159  iocinif  28363  fsumcvg4  28756  ind1a  28842  esumsnf  28885  itg2addnclem2  31994  afvpcfv0  38648
  Copyright terms: Public domain W3C validator