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Theorem cdleme31sn 36503
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 1712 . . . . 5  |-  F/ s  R  .<_  ( P  .\/  Q )
2 nfcsb1v 3436 . . . . 5  |-  F/_ s [_ R  /  s ]_ I
3 nfcsb1v 3436 . . . . 5  |-  F/_ s [_ R  /  s ]_ D
41, 2, 3nfif 3958 . . . 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 4442 . . . 4  |-  ( s  =  R  ->  (
s  .<_  ( P  .\/  Q )  <->  R  .<_  ( P 
.\/  Q ) ) )
7 csbeq1a 3429 . . . 4  |-  ( s  =  R  ->  I  =  [_ R  /  s ]_ I )
8 csbeq1a 3429 . . . 4  |-  ( s  =  R  ->  D  =  [_ R  /  s ]_ D )
96, 7, 8ifbieq12d 3956 . . 3  |-  ( s  =  R  ->  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  D )  =  if ( R 
.<_  ( P  .\/  Q
) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D ) )
105, 9csbiegf 3444 . 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 3832 . 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 2520 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ N  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   F/_wnfc 2602   [_csb 3420   ifcif 3929   class class class wbr 4439  (class class class)co 6270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440
This theorem is referenced by:  cdleme31sn1  36504  cdleme31sn2  36512
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