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Theorem ibllem 22599
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 4438 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  B  <->  0  <_  C ) )
32pm5.32da 645 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_  B )  <->  ( x  e.  A  /\  0  <_  C ) ) )
43ifbid 3937 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 ) )
51adantrr 721 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  C ) )  ->  B  =  C )
65ifeq1da 3945 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
74, 6eqtrd 2470 1  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ifcif 3915   class class class wbr 4426   0cc0 9538    <_ cle 9675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427
This theorem is referenced by:  isibl  22600  isibl2  22601  iblitg  22603  iblcnlem1  22622  iblcnlem  22623  itgcnlem  22624  iblrelem  22625  itgrevallem1  22629  itgeqa  22648
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