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Theorem cdleme31sdnN 30869
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme31sdn.c  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme31sdn.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme31sdn.n  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
Assertion
Ref Expression
cdleme31sdnN  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D
)
Distinct variable groups:    t,  .\/    t, 
./\    t, P    t, Q    t, U    t, W    t,
s
Allowed substitution hints:    C( t, s)    D( t, s)    P( s)    Q( s)    U( s)    I(
t, s)    .\/ ( s)    .<_ ( t, s)    ./\ ( s)    N( t,
s)    W( s)

Proof of Theorem cdleme31sdnN
StepHypRef Expression
1 cdleme31sdn.n . 2  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C
)
2 biid 228 . . 3  |-  ( s 
.<_  ( P  .\/  Q
)  <->  s  .<_  ( P 
.\/  Q ) )
3 vex 2919 . . . 4  |-  s  e. 
_V
4 cdleme31sdn.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
5 cdleme31sdn.c . . . . 5  |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
64, 5cdleme31sc 30866 . . . 4  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  C )
73, 6ax-mp 8 . . 3  |-  [_ s  /  t ]_ D  =  C
82, 7ifbieq2i 3718 . 2  |-  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  C )
91, 8eqtr4i 2427 1  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   [_csb 3211   ifcif 3699   class class class wbr 4172  (class class class)co 6040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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