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Theorem ifeqor 3602
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 3571 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21con3i 127 . . 3  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  -.  ph )
3 iffalse 3572 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
42, 3syl 15 . 2  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  if (
ph ,  A ,  B )  =  B )
54orri 365 1  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1623   ifcif 3565
This theorem is referenced by:  ifpr  3681  muval2  20372  axlowdimlem15  24584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-if 3566
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