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Theorem ifeqor 3973
Description: The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ifeqor  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )

Proof of Theorem ifeqor
StepHypRef Expression
1 iftrue 3935 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21con3i 135 . . 3  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  -.  ph )
32iffalsed 3940 . 2  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  if (
ph ,  A ,  B )  =  B )
43orri 374 1  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 366    = wceq 1398   ifcif 3929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-if 3930
This theorem is referenced by:  ifpr  4064  rabrsn  4086  muval2  23606  relexpxpmin  38226
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