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Theorem elif 3612
Description: Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
Assertion
Ref Expression
elif  |-  ( A  e.  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C )
) )

Proof of Theorem elif
StepHypRef Expression
1 eleq2 2357 . 2  |-  ( if ( ph ,  B ,  C )  =  B  ->  ( A  e.  if ( ph ,  B ,  C )  <->  A  e.  B ) )
2 eleq2 2357 . 2  |-  ( if ( ph ,  B ,  C )  =  C  ->  ( A  e.  if ( ph ,  B ,  C )  <->  A  e.  C ) )
31, 2elimif 3607 1  |-  ( A  e.  if ( ph ,  B ,  C )  <-> 
( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    e. wcel 1696   ifcif 3578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-if 3579
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