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Theorem cdleme31sdnN 29843
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 229 . . 3  |-  ( s 
.<_  ( P  .\/  Q
)  <->  s  .<_  ( P 
.\/  Q ) )
3 vex 2792 . . . 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 29840 . . . 4  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  C )
73, 6ax-mp 10 . . 3  |-  [_ s  /  t ]_ D  =  C
82, 7ifbieq2i 3585 . 2  |-  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  I ,  C )
91, 8eqtr4i 2307 1  |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1628    e. wcel 1688   _Vcvv 2789   [_csb 3082   ifcif 3566   class class class wbr 4024  (class class class)co 5819
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5822
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