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Theorem iftrue 3336
Description: Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
iftrue  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )

Proof of Theorem iftrue
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dedlema 876 . . 3  |-  ( ph  ->  ( x  e.  A  <->  ( ( x  e.  A  /\  ph )  \/  (
x  e.  B  /\  -.  ph ) ) ) )
21abbi2dv 2156 . 2  |-  ( ph  ->  A  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) } )
3 df-if 3332 . 2  |-  if (
ph ,  A ,  B )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
42, 3syl6reqr 2091 1  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    \/ wo 629    = wceq 1243    e. wcel 1393   {cab 2026   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:  iftruei  3337  iftrued  3338  ifbothdc  3357  ifcldcd  3358  fidifsnen  6331  uzin  8505  fzprval  8944  fztpval  8945  expival  9257
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