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Theorem ifeqor 3778
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 3747 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21con3i 130 . . 3  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  -.  ph )
3 iffalse 3748 . . 3  |-  ( -. 
ph  ->  if ( ph ,  A ,  B )  =  B )
42, 3syl 16 . 2  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  if (
ph ,  A ,  B )  =  B )
54orri 367 1  |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 359    = wceq 1653   ifcif 3741
This theorem is referenced by:  ifpr  3858  muval2  20919  axlowdimlem15  25897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-if 3742
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