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Theorem cdleme31se 29838
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 2421 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
2 oveq1 5826 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  T )  =  ( R  .\/  T ) )
32oveq1d 5834 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  T
)  ./\  W )  =  ( ( R 
.\/  T )  ./\  W ) )
43oveq2d 5835 . . . 4  |-  ( s  =  R  ->  ( D  .\/  ( ( s 
.\/  T )  ./\  W ) )  =  ( D  .\/  ( ( R  .\/  T ) 
./\  W ) ) )
54oveq2d 5835 . . 3  |-  ( s  =  R  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
61, 5csbiegf 3122 . 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 3108 . 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 2341 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   [_csb 3082  (class class class)co 5819
This theorem is referenced by:  cdleme31sde  29841  cdleme31sn1c  29844
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  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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