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Theorem cdleme31sn 30642
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 1607 . . . . 5  |-  F/ s  R  .<_  ( P  .\/  Q )
2 nfcsb1v 3115 . . . . 5  |-  F/_ s [_ R  /  s ]_ I
3 nfcsb1v 3115 . . . . 5  |-  F/_ s [_ R  /  s ]_ D
41, 2, 3nfif 3591 . . . 4  |-  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
)
54a1i 10 . . 3  |-  ( R  e.  A  ->  F/_ s if ( R  .<_  ( P 
.\/  Q ) , 
[_ R  /  s ]_ I ,  [_ R  /  s ]_ D
) )
6 breq1 4028 . . . 4  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
7 csbeq1a 3091 . . . 4  |-  ( s  =  R  ->  I  =  [_ R  /  s ]_ I )
8 csbeq1a 3091 . . . 4  |-  ( s  =  R  ->  D  =  [_ R  /  s ]_ D )
96, 7, 8ifbieq12d 3589 . . 3  |-  ( s  =  R  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D ) )
105, 9csbiegf 3123 . 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 3109 . 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 2342 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   F/_wnfc 2408   [_csb 3083   ifcif 3567   class class class wbr 4025  (class class class)co 5860
This theorem is referenced by:  cdleme31sn1  30643  cdleme31sn2  30651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026
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