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Theorem ibllem 19685
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 4249 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  B  <->  0  <_  C ) )
32pm5.32da 624 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_  B )  <->  ( x  e.  A  /\  0  <_  C ) ) )
43ifbid 3781 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 ) )
51adantrr 699 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  C ) )  ->  B  =  C )
65ifeq1da 3788 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
74, 6eqtrd 2474 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 360    = wceq 1653    e. wcel 1727   ifcif 3763   class class class wbr 4237   0cc0 9021    <_ cle 9152
This theorem is referenced by:  isibl  19686  isibl2  19687  iblitg  19689  iblcnlem1  19708  iblcnlem  19709  itgcnlem  19710  iblrelem  19711  itgrevallem1  19715  itgeqa  19734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238
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