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Theorem cdleme31se 29260
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 2386 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
2 oveq1 5717 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  T )  =  ( R  .\/  T ) )
32oveq1d 5725 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  T
)  ./\  W )  =  ( ( R 
.\/  T )  ./\  W ) )
43oveq2d 5726 . . . 4  |-  ( s  =  R  ->  ( D  .\/  ( ( s 
.\/  T )  ./\  W ) )  =  ( D  .\/  ( ( R  .\/  T ) 
./\  W ) ) )
54oveq2d 5726 . . 3  |-  ( s  =  R  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
61, 5csbiegf 3049 . 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 3035 . 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 2310 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   [_csb 3009  (class class class)co 5710
This theorem is referenced by:  cdleme31sde  29263  cdleme31sn1c  29266
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713
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