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Theorem ifeqor 3636
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 3605 . . . 4  |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
21con3i 127 . . 3  |-  ( -.  if ( ph ,  A ,  B )  =  A  ->  -.  ph )
3 iffalse 3606 . . 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 1633   ifcif 3599
This theorem is referenced by:  ifpr  3715  muval2  20425  axlowdimlem15  24970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-if 3600
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