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Theorem cdleme31sn 31114
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 3275 . . . . 5  |-  F/_ s [_ R  /  s ]_ I
3 nfcsb1v 3275 . . . . 5  |-  F/_ s [_ R  /  s ]_ D
41, 2, 3nfif 3755 . . . 4  |-  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
54a1i 11 . . 3  |-  ( R  e.  A  ->  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
) )
6 breq1 4207 . . . 4  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
7 csbeq1a 3251 . . . 4  |-  ( s  =  R  ->  I  =  [_ R  /  s ]_ I )
8 csbeq1a 3251 . . . 4  |-  ( s  =  R  ->  D  =  [_ R  /  s ]_ D )
96, 7, 8ifbieq12d 3753 . . 3  |-  ( s  =  R  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D ) )
105, 9csbiegf 3283 . 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 3269 . 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 2492 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   F/_wnfc 2558   [_csb 3243   ifcif 3731   class class class wbr 4204  (class class class)co 6073
This theorem is referenced by:  cdleme31sn1  31115  cdleme31sn2  31123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205
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