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Theorem cdleme31se2 29839
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 2420 . . . . 5  |-  F/_ t
( P  .\/  Q
)
2 nfcv 2420 . . . . 5  |-  F/_ t  ./\
3 nfcsb1v 3114 . . . . . 6  |-  F/_ t [_ S  /  t ]_ D
4 nfcv 2420 . . . . . 6  |-  F/_ t  .\/
5 nfcv 2420 . . . . . 6  |-  F/_ t
( ( R  .\/  S )  ./\  W )
63, 4, 5nfov 5842 . . . . 5  |-  F/_ t
( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) )
71, 2, 6nfov 5842 . . . 4  |-  F/_ t
( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) )
87a1i 12 . . 3  |-  ( S  e.  A  ->  F/_ t
( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
9 csbeq1a 3090 . . . . 5  |-  ( t  =  S  ->  D  =  [_ S  /  t ]_ D )
10 oveq2 5827 . . . . . 6  |-  ( t  =  S  ->  ( R  .\/  t )  =  ( R  .\/  S
) )
1110oveq1d 5834 . . . . 5  |-  ( t  =  S  ->  (
( R  .\/  t
)  ./\  W )  =  ( ( R 
.\/  S )  ./\  W ) )
129, 11oveq12d 5837 . . . 4  |-  ( t  =  S  ->  ( D  .\/  ( ( R 
.\/  t )  ./\  W ) )  =  (
[_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) ) )
1312oveq2d 5835 . . 3  |-  ( t  =  S  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
148, 13csbiegf 3122 . 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 3108 . 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 2341 1  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   F/_wnfc 2407   [_csb 3082  (class class class)co 5819
This theorem is referenced by:  cdlemeg47rv2  29966
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-ral 2549  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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