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Theorem cdleme31snd 29742
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 5817 . . . . 5  |-  ( ( v  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )  e. 
_V
31, 2eqeltri 2328 . . . 4  |-  N  e. 
_V
43ax-gen 1536 . . 3  |-  A. v  N  e.  _V
5 csbnestgOLD 3107 . . 3  |-  ( ( S  e.  A  /\  A. v  N  e.  _V )  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  [_ [_ S  /  v ]_ N  /  t ]_ D
)
64, 5mpan2 655 . 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 29740 . . . 4  |-  ( S  e.  A  ->  [_ S  /  v ]_ N  =  O )
98csbeq1d 3062 . . 3  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  [_ O  /  t ]_ D )
10 ovex 5817 . . . . 5  |-  ( ( S  .\/  V ) 
./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
) )  e.  _V
117, 10eqeltri 2328 . . . 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 29740 . . . 4  |-  ( O  e.  _V  ->  [_ O  /  t ]_ D  =  E )
1511, 14ax-mp 10 . . 3  |-  [_ O  /  t ]_ D  =  E
169, 15syl6eq 2306 . 2  |-  ( S  e.  A  ->  [_ [_ S  /  v ]_ N  /  t ]_ D  =  E )
176, 16eqtrd 2290 1  |-  ( S  e.  A  ->  [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532    = wceq 1619    e. wcel 1621   _Vcvv 2763   [_csb 3056  (class class class)co 5792
This theorem is referenced by:  cdlemeg46ngfr  29874
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-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  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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