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Theorem cdleme31sn 29828
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31sn.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
cdleme31sn.c  |-  C  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
)
Assertion
Ref Expression
cdleme31sn  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  C )
Distinct variable groups:    A, s    .\/ , s    .<_ , s    P, s    Q, s    R, s
Allowed substitution hints:    C( s)    D( s)    I( s)    N( s)

Proof of Theorem cdleme31sn
StepHypRef Expression
1 nfv 1629 . . . . 5  |-  F/ s  R  .<_  ( P  .\/  Q )
2 nfcsb1v 3088 . . . . 5  |-  F/_ s [_ R  /  s ]_ I
3 nfcsb1v 3088 . . . . 5  |-  F/_ s [_ R  /  s ]_ D
41, 2, 3nfif 3564 . . . 4  |-  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
54a1i 12 . . 3  |-  ( R  e.  A  ->  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
) )
6 breq1 4001 . . . 4  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
7 csbeq1a 3064 . . . 4  |-  ( s  =  R  ->  I  =  [_ R  /  s ]_ I )
8 csbeq1a 3064 . . . 4  |-  ( s  =  R  ->  D  =  [_ R  /  s ]_ D )
96, 7, 8ifbieq12d 3562 . . 3  |-  ( s  =  R  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D ) )
105, 9csbiegf 3096 . 2  |-  ( R  e.  A  ->  [_ R  /  s ]_ if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D ) )
11 cdleme31sn.n . . 3  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
1211csbeq2i 3082 . 2  |-  [_ R  /  s ]_ N  =  [_ R  /  s ]_ if ( s  .<_  ( P  .\/  Q ) ,  I ,  D
)
13 cdleme31sn.c . 2  |-  C  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  / 
s ]_ I ,  [_ R  /  s ]_ D
)
1410, 12, 133eqtr4g 2315 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   F/_wnfc 2381   [_csb 3056   ifcif 3540   class class class wbr 3998  (class class class)co 5793
This theorem is referenced by:  cdleme31sn1  29829  cdleme31sn2  29837
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3541  df-sn 3621  df-pr 3622  df-op 3624  df-br 3999
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