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

Theorem ifeqor 3936
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 3900 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21con3i 135 . . 3  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  -.  ph )
3 iffalse 3902 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
42, 3syl 16 . 2  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  if (
ph ,  A ,  B )  =  B )
54orri 376 1  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1370   ifcif 3894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-if 3895
This theorem is referenced by:  ifpr  4027  rabrsn  4048  muval2  22600  axlowdimlem15  23349
  Copyright terms: Public domain W3C validator