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Theorem cdleme31se2 29739
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 2394 . . . . 5  |-  F/_ t
( P  .\/  Q
)
2 nfcv 2394 . . . . 5  |-  F/_ t  ./\
3 nfcsb1v 3088 . . . . . 6  |-  F/_ t [_ S  /  t ]_ D
4 nfcv 2394 . . . . . 6  |-  F/_ t  .\/
5 nfcv 2394 . . . . . 6  |-  F/_ t
( ( R  .\/  S )  ./\  W )
63, 4, 5nfov 5815 . . . . 5  |-  F/_ t
( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) )
71, 2, 6nfov 5815 . . . 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 3064 . . . . 5  |-  ( t  =  S  ->  D  =  [_ S  /  t ]_ D )
10 oveq2 5800 . . . . . 6  |-  ( t  =  S  ->  ( R  .\/  t )  =  ( R  .\/  S
) )
1110oveq1d 5807 . . . . 5  |-  ( t  =  S  ->  (
( R  .\/  t
)  ./\  W )  =  ( ( R 
.\/  S )  ./\  W ) )
129, 11oveq12d 5810 . . . 4  |-  ( t  =  S  ->  ( D  .\/  ( ( R 
.\/  t )  ./\  W ) )  =  (
[_ S  /  t ]_ D  .\/  ( ( R  .\/  S ) 
./\  W ) ) )
1312oveq2d 5808 . . 3  |-  ( t  =  S  ->  (
( P  .\/  Q
)  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W
) ) )  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
) ) )
148, 13csbiegf 3096 . 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 3082 . 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 2315 1  |-  ( S  e.  A  ->  [_ S  /  t ]_ E  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   F/_wnfc 2381   [_csb 3056  (class class class)co 5792
This theorem is referenced by:  cdlemeg47rv2  29866
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689  df-ov 5795
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