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Theorem cdleme31snd 34339
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 csbnestg 3795 . 2  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  [_ [_ S  / 
v ]_ N  /  t ]_ D )
2 cdleme31snd.n . . . . 5  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
3 cdleme31snd.o . . . . 5  |-  O  =  ( ( S  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )
42, 3cdleme31sc 34337 . . . 4  |-  ( S  e.  A  ->  [_ S  /  v ]_ N  =  O )
54csbeq1d 3396 . . 3  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  [_ O  /  t ]_ D )
6 ovex 6218 . . . . 5  |-  ( ( S  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )  e.  _V
73, 6eqeltri 2535 . . . 4  |-  O  e. 
_V
8 cdleme31snd.d . . . . 5  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
9 cdleme31snd.e . . . . 5  |-  E  =  ( ( O  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  O )  ./\  W )
) )
108, 9cdleme31sc 34337 . . . 4  |-  ( O  e.  _V  ->  [_ O  /  t ]_ D  =  E )
117, 10ax-mp 5 . . 3  |-  [_ O  /  t ]_ D  =  E
125, 11syl6eq 2508 . 2  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  E )
131, 12eqtrd 2492 1  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3071   [_csb 3389  (class class class)co 6193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196
This theorem is referenced by:  cdlemeg46ngfr  34471
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