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Theorem br4 24186
Description: Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
Hypotheses
Ref Expression
br4.1  |-  ( a  =  A  ->  ( ph 
<->  ps ) )
br4.2  |-  ( b  =  B  ->  ( ps 
<->  ch ) )
br4.3  |-  ( c  =  C  ->  ( ch 
<->  th ) )
br4.4  |-  ( d  =  D  ->  ( th 
<->  ta ) )
br4.5  |-  ( x  =  X  ->  P  =  Q )
br4.6  |-  R  =  { <. p ,  q
>.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) }
Assertion
Ref Expression
br4  |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  -> 
( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
Distinct variable groups:    a, b,
c, d, p, q, x, A    B, a,
b, c, d, p, q, x    ch, b    Q, a, b, c, d, x    C, a, b, c, d, p, q, x    D, a, b, c, d, p, q, x    ps, a    X, a, b, c, d, x    P, a, b, c, d, p, q    S, a, b, c, d, p, q, x    ta, a, b, c, d, x    th, c    ph, p, q, x
Allowed substitution hints:    ph( a, b, c, d)    ps( x, q, p, b, c, d)    ch( x, q, p, a, c, d)    th( x, q, p, a, b, d)    ta( q, p)    P( x)    Q( q, p)    R( x, q, p, a, b, c, d)    X( q, p)

Proof of Theorem br4
StepHypRef Expression
1 opex 4253 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4253 . . 3  |-  <. C ,  D >.  e.  _V
3 eqeq1 2302 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( p  = 
<. a ,  b >.  <->  <. A ,  B >.  = 
<. a ,  b >.
) )
433anbi1d 1256 . . . . . 6  |-  ( p  =  <. A ,  B >.  ->  ( ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  ( <. A ,  B >.  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) ) )
54rexbidv 2577 . . . . 5  |-  ( p  =  <. A ,  B >.  ->  ( E. d  e.  P  ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) ) )
652rexbidv 2599 . . . 4  |-  ( p  =  <. A ,  B >.  ->  ( E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) ) )
762rexbidv 2599 . . 3  |-  ( p  =  <. A ,  B >.  ->  ( E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) ) )
8 eqeq1 2302 . . . . . . 7  |-  ( q  =  <. C ,  D >.  ->  ( q  = 
<. c ,  d >.  <->  <. C ,  D >.  = 
<. c ,  d >.
) )
983anbi2d 1257 . . . . . 6  |-  ( q  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) ) )
109rexbidv 2577 . . . . 5  |-  ( q  =  <. C ,  D >.  ->  ( E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  q  =  <. c ,  d
>.  /\  ph )  <->  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
11102rexbidv 2599 . . . 4  |-  ( q  =  <. C ,  D >.  ->  ( E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  q  =  <. c ,  d
>.  /\  ph )  <->  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
12112rexbidv 2599 . . 3  |-  ( q  =  <. C ,  D >.  ->  ( E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  q  =  <. c ,  d
>.  /\  ph )  <->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
13 br4.6 . . 3  |-  R  =  { <. p ,  q
>.  |  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) }
141, 2, 7, 12, 13brab 4303 . 2  |-  ( <. A ,  B >. R
<. C ,  D >.  <->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) )
15 vex 2804 . . . . . . . . . . . 12  |-  a  e. 
_V
16 vex 2804 . . . . . . . . . . . 12  |-  b  e. 
_V
1715, 16opth 4261 . . . . . . . . . . 11  |-  ( <.
a ,  b >.  =  <. A ,  B >.  <-> 
( a  =  A  /\  b  =  B ) )
18 br4.1 . . . . . . . . . . . 12  |-  ( a  =  A  ->  ( ph 
<->  ps ) )
19 br4.2 . . . . . . . . . . . 12  |-  ( b  =  B  ->  ( ps 
<->  ch ) )
2018, 19sylan9bb 680 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ph  <->  ch )
)
2117, 20sylbi 187 . . . . . . . . . 10  |-  ( <.
a ,  b >.  =  <. A ,  B >.  ->  ( ph  <->  ch )
)
2221eqcoms 2299 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. a ,  b >.  ->  ( ph  <->  ch )
)
23 vex 2804 . . . . . . . . . . . 12  |-  c  e. 
_V
24 vex 2804 . . . . . . . . . . . 12  |-  d  e. 
_V
2523, 24opth 4261 . . . . . . . . . . 11  |-  ( <.
c ,  d >.  =  <. C ,  D >.  <-> 
( c  =  C  /\  d  =  D ) )
26 br4.3 . . . . . . . . . . . 12  |-  ( c  =  C  ->  ( ch 
<->  th ) )
27 br4.4 . . . . . . . . . . . 12  |-  ( d  =  D  ->  ( th 
<->  ta ) )
2826, 27sylan9bb 680 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ch  <->  ta )
)
2925, 28sylbi 187 . . . . . . . . . 10  |-  ( <.
c ,  d >.  =  <. C ,  D >.  ->  ( ch  <->  ta )
)
3029eqcoms 2299 . . . . . . . . 9  |-  ( <. C ,  D >.  = 
<. c ,  d >.  ->  ( ch  <->  ta )
)
3122, 30sylan9bb 680 . . . . . . . 8  |-  ( (
<. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>. )  ->  ( ph  <->  ta ) )
3231biimp3a 1281 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  ->  ta )
3332a1i 10 . . . . . 6  |-  ( ( ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q
) )  /\  (
x  e.  S  /\  a  e.  P )
)  /\  ( b  e.  P  /\  c  e.  P ) )  /\  d  e.  P )  ->  ( ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph )  ->  ta ) )
3433rexlimdva 2680 . . . . 5  |-  ( ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q )  /\  ( C  e.  Q  /\  D  e.  Q )
)  /\  ( x  e.  S  /\  a  e.  P ) )  /\  ( b  e.  P  /\  c  e.  P
) )  ->  ( E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  ->  ta ) )
3534rexlimdvva 2687 . . . 4  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ( x  e.  S  /\  a  e.  P
) )  ->  ( E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  ->  ta ) )
3635rexlimdvva 2687 . . 3  |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  -> 
( E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph )  ->  ta ) )
37 simpl1 958 . . . . 5  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  X  e.  S )
38 simpl2l 1008 . . . . . 6  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  A  e.  Q )
39 simpl2r 1009 . . . . . 6  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  B  e.  Q )
40 simpl3l 1010 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  C  e.  Q )
41 simpl3r 1011 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  D  e.  Q )
42 eqidd 2297 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  <. A ,  B >.  =  <. A ,  B >. )
43 eqidd 2297 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  <. C ,  D >.  =  <. C ,  D >. )
44 simpr 447 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  ta )
45 opeq1 3812 . . . . . . . . . 10  |-  ( c  =  C  ->  <. c ,  d >.  =  <. C ,  d >. )
4645eqeq2d 2307 . . . . . . . . 9  |-  ( c  =  C  ->  ( <. C ,  D >.  = 
<. c ,  d >.  <->  <. C ,  D >.  = 
<. C ,  d >.
) )
4746, 263anbi23d 1255 . . . . . . . 8  |-  ( c  =  C  ->  (
( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ch )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  d >.  /\  th ) ) )
48 opeq2 3813 . . . . . . . . . 10  |-  ( d  =  D  ->  <. C , 
d >.  =  <. C ,  D >. )
4948eqeq2d 2307 . . . . . . . . 9  |-  ( d  =  D  ->  ( <. C ,  D >.  = 
<. C ,  d >.  <->  <. C ,  D >.  = 
<. C ,  D >. ) )
5049, 273anbi23d 1255 . . . . . . . 8  |-  ( d  =  D  ->  (
( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  d
>.  /\  th )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  D >.  /\  ta ) ) )
5147, 50rspc2ev 2905 . . . . . . 7  |-  ( ( C  e.  Q  /\  D  e.  Q  /\  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  D >.  /\  ta ) )  ->  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ch ) )
5240, 41, 42, 43, 44, 51syl113anc 1194 . . . . . 6  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ch ) )
53 opeq1 3812 . . . . . . . . . 10  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
5453eqeq2d 2307 . . . . . . . . 9  |-  ( a  =  A  ->  ( <. A ,  B >.  = 
<. a ,  b >.  <->  <. A ,  B >.  = 
<. A ,  b >.
) )
5554, 183anbi13d 1254 . . . . . . . 8  |-  ( a  =  A  ->  (
( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  ( <. A ,  B >.  =  <. A ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ps ) ) )
56552rexbidv 2599 . . . . . . 7  |-  ( a  =  A  ->  ( E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. A ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ps ) ) )
57 opeq2 3813 . . . . . . . . . 10  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
5857eqeq2d 2307 . . . . . . . . 9  |-  ( b  =  B  ->  ( <. A ,  B >.  = 
<. A ,  b >.  <->  <. A ,  B >.  = 
<. A ,  B >. ) )
5958, 193anbi13d 1254 . . . . . . . 8  |-  ( b  =  B  ->  (
( <. A ,  B >.  =  <. A ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ps )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ch ) ) )
60592rexbidv 2599 . . . . . . 7  |-  ( b  =  B  ->  ( E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. A ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ps )  <->  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ch ) ) )
6156, 60rspc2ev 2905 . . . . . 6  |-  ( ( A  e.  Q  /\  B  e.  Q  /\  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  = 
<. A ,  B >.  /\ 
<. C ,  D >.  = 
<. c ,  d >.  /\  ch ) )  ->  E. a  e.  Q  E. b  e.  Q  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) )
6238, 39, 52, 61syl3anc 1182 . . . . 5  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  E. a  e.  Q  E. b  e.  Q  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) )
63 br4.5 . . . . . . 7  |-  ( x  =  X  ->  P  =  Q )
6463rexeqdv 2756 . . . . . . . . 9  |-  ( x  =  X  ->  ( E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
6563, 64rexeqbidv 2762 . . . . . . . 8  |-  ( x  =  X  ->  ( E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
6663, 65rexeqbidv 2762 . . . . . . 7  |-  ( x  =  X  ->  ( E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  E. b  e.  Q  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
6763, 66rexeqbidv 2762 . . . . . 6  |-  ( x  =  X  ->  ( E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  E. a  e.  Q  E. b  e.  Q  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
6867rspcev 2897 . . . . 5  |-  ( ( X  e.  S  /\  E. a  e.  Q  E. b  e.  Q  E. c  e.  Q  E. d  e.  Q  ( <. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) )  ->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) )
6937, 62, 68syl2anc 642 . . . 4  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) )
7069ex 423 . . 3  |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  -> 
( ta  ->  E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) ) )
7136, 70impbid 183 . 2  |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  -> 
( E. x  e.  S  E. a  e.  P  E. b  e.  P  E. c  e.  P  E. d  e.  P  ( <. A ,  B >.  =  <. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ph ) 
<->  ta ) )
7214, 71syl5bb 248 1  |-  ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  -> 
( <. A ,  B >. R <. C ,  D >.  <->  ta ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039   {copab 4092
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-mo 2161  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-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-br 4040  df-opab 4094
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