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Theorem cdleme31se2 31194
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)
Hypotheses
Ref Expression
cdleme31se2.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )
cdleme31se2.y  |-  Y  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme31se2  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  Y )
Distinct variable groups:    t, A    t, 
.\/    t,  ./\    t, P    t, Q    t, R    t, S    t, W
Allowed substitution hints:    D( t)    E( t)    Y( t)

Proof of Theorem cdleme31se2
StepHypRef Expression
1 nfcv 2432 . . . . 5  |-  F/_ t
( P  .\/  Q
)
2 nfcv 2432 . . . . 5  |-  F/_ t  ./\
3 nfcsb1v 3126 . . . . . 6  |-  F/_ t [_ S  /  t ]_ D
4 nfcv 2432 . . . . . 6  |-  F/_ t  .\/
5 nfcv 2432 . . . . . 6  |-  F/_ t
( ( R  .\/  S )  ./\  W )
63, 4, 5nfov 5897 . . . . 5  |-  F/_ t
( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) )
71, 2, 6nfov 5897 . . . 4  |-  F/_ t
( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) )
87a1i 10 . . 3  |-  ( S  e.  A  ->  F/_ t
( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
9 csbeq1a 3102 . . . . 5  |-  ( t  =  S  ->  D  =  [_ S  /  t ]_ D )
10 oveq2 5882 . . . . . 6  |-  ( t  =  S  ->  ( R  .\/  t )  =  ( R  .\/  S
) )
1110oveq1d 5889 . . . . 5  |-  ( t  =  S  ->  (
( R  .\/  t
)  ./\  W )  =  ( ( R 
.\/  S )  ./\  W ) )
129, 11oveq12d 5892 . . . 4  |-  ( t  =  S  ->  ( D  .\/  ( ( R 
.\/  t )  ./\  W ) )  =  (
[_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) ) )
1312oveq2d 5890 . . 3  |-  ( t  =  S  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
148, 13csbiegf 3134 . 2  |-  ( S  e.  A  ->  [_ S  /  t ]_ (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
15 cdleme31se2.e . . 3  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )
1615csbeq2i 3120 . 2  |-  [_ S  /  t ]_ E  =  [_ S  /  t ]_ ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )
17 cdleme31se2.y . 2  |-  Y  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) )
1814, 16, 173eqtr4g 2353 1  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   F/_wnfc 2419   [_csb 3094  (class class class)co 5874
This theorem is referenced by:  cdlemeg47rv2  31321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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