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Theorem cdleme31se 33861
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 2570 . . 3  |-  ( R  e.  A  ->  F/_ s
( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
2 oveq1 6256 . . . . . 6  |-  ( s  =  R  ->  (
s  .\/  T )  =  ( R  .\/  T ) )
32oveq1d 6264 . . . . 5  |-  ( s  =  R  ->  (
( s  .\/  T
)  ./\  W )  =  ( ( R 
.\/  T )  ./\  W ) )
43oveq2d 6265 . . . 4  |-  ( s  =  R  ->  ( D  .\/  ( ( s 
.\/  T )  ./\  W ) )  =  ( D  .\/  ( ( R  .\/  T ) 
./\  W ) ) )
54oveq2d 6265 . . 3  |-  ( s  =  R  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
) )  =  ( ( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
) ) )
61, 5csbiegf 3362 . 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 3755 . 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 2487 1  |-  ( R  e.  A  ->  [_ R  /  s ]_ E  =  Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   [_csb 3338  (class class class)co 6249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-ov 6252
This theorem is referenced by:  cdleme31sde  33864  cdleme31sn1c  33867
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