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Theorem br4 25329
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 4387 . . 3  |-  <. A ,  B >.  e.  _V
2 opex 4387 . . 3  |-  <. C ,  D >.  e.  _V
3 eqeq1 2410 . . . . . . 7  |-  ( p  =  <. A ,  B >.  ->  ( p  = 
<. a ,  b >.  <->  <. A ,  B >.  = 
<. a ,  b >.
) )
433anbi1d 1258 . . . . . 6  |-  ( p  =  <. A ,  B >.  ->  ( ( p  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  ( <. A ,  B >.  =  <. a ,  b
>.  /\  q  =  <. c ,  d >.  /\  ph ) ) )
54rexbidv 2687 . . . . 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 2709 . . . 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 2709 . . 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 2410 . . . . . . 7  |-  ( q  =  <. C ,  D >.  ->  ( q  = 
<. c ,  d >.  <->  <. C ,  D >.  = 
<. c ,  d >.
) )
983anbi2d 1259 . . . . . 6  |-  ( q  =  <. C ,  D >.  ->  ( ( <. A ,  B >.  = 
<. a ,  b >.  /\  q  =  <. c ,  d >.  /\  ph ) 
<->  ( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph ) ) )
109rexbidv 2687 . . . . 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 2709 . . . 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 2709 . . 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 4437 . 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 2919 . . . . . . . . . . . 12  |-  a  e. 
_V
16 vex 2919 . . . . . . . . . . . 12  |-  b  e. 
_V
1715, 16opth 4395 . . . . . . . . . . 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 681 . . . . . . . . . . 11  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ph  <->  ch )
)
2117, 20sylbi 188 . . . . . . . . . 10  |-  ( <.
a ,  b >.  =  <. A ,  B >.  ->  ( ph  <->  ch )
)
2221eqcoms 2407 . . . . . . . . 9  |-  ( <. A ,  B >.  = 
<. a ,  b >.  ->  ( ph  <->  ch )
)
23 vex 2919 . . . . . . . . . . . 12  |-  c  e. 
_V
24 vex 2919 . . . . . . . . . . . 12  |-  d  e. 
_V
2523, 24opth 4395 . . . . . . . . . . 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 681 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  d  =  D )  ->  ( ch  <->  ta )
)
2925, 28sylbi 188 . . . . . . . . . 10  |-  ( <.
c ,  d >.  =  <. C ,  D >.  ->  ( ch  <->  ta )
)
3029eqcoms 2407 . . . . . . . . 9  |-  ( <. C ,  D >.  = 
<. c ,  d >.  ->  ( ch  <->  ta )
)
3122, 30sylan9bb 681 . . . . . . . 8  |-  ( (
<. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>. )  ->  ( ph  <->  ta ) )
3231biimp3a 1283 . . . . . . 7  |-  ( (
<. A ,  B >.  = 
<. a ,  b >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  ->  ta )
3332a1i 11 . . . . . 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 2790 . . . . 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 2797 . . . 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 2797 . . 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 960 . . . . 5  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  X  e.  S )
38 simpl2l 1010 . . . . . 6  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  A  e.  Q )
39 simpl2r 1011 . . . . . 6  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  B  e.  Q )
40 simpl3l 1012 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  C  e.  Q )
41 simpl3r 1013 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  D  e.  Q )
42 eqidd 2405 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  <. A ,  B >.  =  <. A ,  B >. )
43 eqidd 2405 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  <. C ,  D >.  =  <. C ,  D >. )
44 simpr 448 . . . . . . 7  |-  ( ( ( X  e.  S  /\  ( A  e.  Q  /\  B  e.  Q
)  /\  ( C  e.  Q  /\  D  e.  Q ) )  /\  ta )  ->  ta )
45 opeq1 3944 . . . . . . . . . 10  |-  ( c  =  C  ->  <. c ,  d >.  =  <. C ,  d >. )
4645eqeq2d 2415 . . . . . . . . 9  |-  ( c  =  C  ->  ( <. C ,  D >.  = 
<. c ,  d >.  <->  <. C ,  D >.  = 
<. C ,  d >.
) )
4746, 263anbi23d 1257 . . . . . . . 8  |-  ( c  =  C  ->  (
( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ch )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  d >.  /\  th ) ) )
48 opeq2 3945 . . . . . . . . . 10  |-  ( d  =  D  ->  <. C , 
d >.  =  <. C ,  D >. )
4948eqeq2d 2415 . . . . . . . . 9  |-  ( d  =  D  ->  ( <. C ,  D >.  = 
<. C ,  d >.  <->  <. C ,  D >.  = 
<. C ,  D >. ) )
5049, 273anbi23d 1257 . . . . . . . 8  |-  ( d  =  D  ->  (
( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  d
>.  /\  th )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. C ,  D >.  /\  ta ) ) )
5147, 50rspc2ev 3020 . . . . . . 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 1196 . . . . . 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 3944 . . . . . . . . . 10  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
5453eqeq2d 2415 . . . . . . . . 9  |-  ( a  =  A  ->  ( <. A ,  B >.  = 
<. a ,  b >.  <->  <. A ,  B >.  = 
<. A ,  b >.
) )
5554, 183anbi13d 1256 . . . . . . . 8  |-  ( a  =  A  ->  (
( <. A ,  B >.  =  <. a ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ph )  <->  ( <. A ,  B >.  =  <. A ,  b >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ps ) ) )
56552rexbidv 2709 . . . . . . 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 3945 . . . . . . . . . 10  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
5857eqeq2d 2415 . . . . . . . . 9  |-  ( b  =  B  ->  ( <. A ,  B >.  = 
<. A ,  b >.  <->  <. A ,  B >.  = 
<. A ,  B >. ) )
5958, 193anbi13d 1256 . . . . . . . 8  |-  ( b  =  B  ->  (
( <. A ,  B >.  =  <. A ,  b
>.  /\  <. C ,  D >.  =  <. c ,  d
>.  /\  ps )  <->  ( <. A ,  B >.  =  <. A ,  B >.  /\  <. C ,  D >.  =  <. c ,  d >.  /\  ch ) ) )
60592rexbidv 2709 . . . . . . 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 3020 . . . . . 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 1184 . . . . 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 2871 . . . . . . . . 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 2877 . . . . . . . 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 2877 . . . . . . 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 2877 . . . . . 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 3012 . . . . 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 643 . . . 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 424 . . 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 184 . 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 249 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   <.cop 3777   class class class wbr 4172   {copab 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227
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