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Theorem cdleme31snd 31197
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)
Hypotheses
Ref Expression
cdleme31snd.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme31snd.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdleme31snd.e  |-  E  =  ( ( O  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  O )  ./\  W )
) )
cdleme31snd.o  |-  O  =  ( ( S  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme31snd  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
Distinct variable groups:    v, A    v, D    v, t,  .\/    t, 
./\ , v    t, O    t, P, v    t, Q, v   
v, S    t, U, v    v, V    t, W, v
Allowed substitution hints:    A( t)    D( t)    S( t)    E( v, t)    N( v, t)    O( v)    V( t)

Proof of Theorem cdleme31snd
StepHypRef Expression
1 cdleme31snd.n . . . . 5  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
2 ovex 5899 . . . . 5  |-  ( ( v  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  e. 
_V
31, 2eqeltri 2366 . . . 4  |-  N  e. 
_V
43ax-gen 1536 . . 3  |-  A. v  N  e.  _V
5 csbnestgOLD 3145 . . 3  |-  ( ( S  e.  A  /\  A. v  N  e.  _V )  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  [_ [_ S  /  v ]_ N  /  t ]_ D
)
64, 5mpan2 652 . 2  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  [_ [_ S  / 
v ]_ N  /  t ]_ D )
7 cdleme31snd.o . . . . 5  |-  O  =  ( ( S  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )
81, 7cdleme31sc 31195 . . . 4  |-  ( S  e.  A  ->  [_ S  /  v ]_ N  =  O )
98csbeq1d 3100 . . 3  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  [_ O  /  t ]_ D )
10 ovex 5899 . . . . 5  |-  ( ( S  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )  e.  _V
117, 10eqeltri 2366 . . . 4  |-  O  e. 
_V
12 cdleme31snd.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
13 cdleme31snd.e . . . . 5  |-  E  =  ( ( O  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  O )  ./\  W )
) )
1412, 13cdleme31sc 31195 . . . 4  |-  ( O  e.  _V  ->  [_ O  /  t ]_ D  =  E )
1511, 14ax-mp 8 . . 3  |-  [_ O  /  t ]_ D  =  E
169, 15syl6eq 2344 . 2  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  E )
176, 16eqtrd 2328 1  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094  (class class class)co 5874
This theorem is referenced by:  cdlemeg46ngfr  31329
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  ax-nul 4165
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-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  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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