MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ibllem Structured version   Unicode version

Theorem ibllem 21934
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 4459 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
0  <_  B  <->  0  <_  C ) )
32pm5.32da 641 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  0  <_  B )  <->  ( x  e.  A  /\  0  <_  C ) ) )
43ifbid 3961 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  B ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 ) )
51adantrr 716 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  0  <_  C ) )  ->  B  =  C )
65ifeq1da 3969 . 2  |-  ( ph  ->  if ( ( x  e.  A  /\  0  <_  C ) ,  B ,  0 )  =  if ( ( x  e.  A  /\  0  <_  C ) ,  C ,  0 ) )
74, 6eqtrd 2508 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 369    = wceq 1379    e. wcel 1767   ifcif 3939   class class class wbr 4447   0cc0 9492    <_ cle 9629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448
This theorem is referenced by:  isibl  21935  isibl2  21936  iblitg  21938  iblcnlem1  21957  iblcnlem  21958  itgcnlem  21959  iblrelem  21960  itgrevallem1  21964  itgeqa  21983
  Copyright terms: Public domain W3C validator