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Theorem ibllem 19119
Description: Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
Hypothesis
Ref Expression
ibllem.1  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
Assertion
Ref Expression
ibllem  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )

Proof of Theorem ibllem
StepHypRef Expression
1 ibllem.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  B  =  C )
21breq2d 4035 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  B  <->  0  <_  C ) )
32pm5.32da 622 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_  B )  <->  ( x  e.  A  /\  0  <_  C ) ) )
43ifbid 3583 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 ) )
51adantrr 697 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  C ) )  ->  B  =  C )
65ifeq1da 3590 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
74, 6eqtrd 2315 1  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023   0cc0 8737    <_ cle 8868
This theorem is referenced by:  isibl  19120  isibl2  19121  iblitg  19123  iblcnlem1  19142  iblcnlem  19143  itgcnlem  19144  iblrelem  19145  itgrevallem1  19149  itgeqa  19168
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-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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