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Theorem cdleme31snd 31022
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 3293 . 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 31020 . . . 4  |-  ( S  e.  A  ->  [_ S  /  v ]_ N  =  O )
54csbeq1d 3249 . . 3  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  [_ O  /  t ]_ D )
6 ovex 6097 . . . . 5  |-  ( ( S  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )  e.  _V
73, 6eqeltri 2505 . . . 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 31020 . . . 4  |-  ( O  e.  _V  ->  [_ O  /  t ]_ D  =  E )
117, 10ax-mp 8 . . 3  |-  [_ O  /  t ]_ D  =  E
125, 11syl6eq 2483 . 2  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  E )
131, 12eqtrd 2467 1  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243  (class class class)co 6072
This theorem is referenced by:  cdlemeg46ngfr  31154
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5409  df-fv 5453  df-ov 6075
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