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Theorem ibllem 19135
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 4051 . . . 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 3596 . 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 3603 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
74, 6eqtrd 2328 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 1632    e. wcel 1696   ifcif 3578   class class class wbr 4039   0cc0 8753    <_ cle 8884
This theorem is referenced by:  isibl  19136  isibl2  19137  iblitg  19139  iblcnlem1  19158  iblcnlem  19159  itgcnlem  19160  iblrelem  19161  itgrevallem1  19165  itgeqa  19184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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