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Theorem cdleme31se 30571
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
Hypotheses
Ref Expression
cdleme31se.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )
cdleme31se.y  |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) )
Assertion
Ref Expression
cdleme31se  |-  ( R  e.  A  ->  [_ R  /  s ]_ E  =  Y )
Distinct variable groups:    A, s    D, s    .\/ , s    ./\ , s    P, s    Q, s    R, s    W, s    T, s
Allowed substitution hints:    E( s)    Y( s)

Proof of Theorem cdleme31se
StepHypRef Expression
1 nfcvd 2420 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
2 oveq1 5865 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  T )  =  ( R  .\/  T ) )
32oveq1d 5873 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  T
)  ./\  W )  =  ( ( R 
.\/  T )  ./\  W ) )
43oveq2d 5874 . . . 4  |-  ( s  =  R  ->  ( D  .\/  ( ( s 
.\/  T )  ./\  W ) )  =  ( D  .\/  ( ( R  .\/  T ) 
./\  W ) ) )
54oveq2d 5874 . . 3  |-  ( s  =  R  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
61, 5csbiegf 3121 . 2  |-  ( R  e.  A  ->  [_ R  /  s ]_ (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
7 cdleme31se.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )
87csbeq2i 3107 . 2  |-  [_ R  /  s ]_ E  =  [_ R  /  s ]_ ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )
9 cdleme31se.y . 2  |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) )
106, 8, 93eqtr4g 2340 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   [_csb 3081  (class class class)co 5858
This theorem is referenced by:  cdleme31sde  30574  cdleme31sn1c  30577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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