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Theorem th3qlem1 6650
Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60. The third hypothesis is the compatibility assumption. (Contributed by NM, 3-Aug-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
th3qlem1.1  |-  .~  Er  S
th3qlem1.3  |-  ( ( ( y  e.  S  /\  w  e.  S
)  /\  ( z  e.  S  /\  v  e.  S ) )  -> 
( ( y  .~  w  /\  z  .~  v
)  ->  ( y  .+  z )  .~  (
w  .+  v )
) )
Assertion
Ref Expression
th3qlem1  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  ->  E* x E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )
)
Distinct variable groups:    x, y,
z, w, v,  .+    x, 
.~ , y, z, w, v    x, S, y, z, w, v    x, A, y, z, w, v   
x, B, y, z, w, v

Proof of Theorem th3qlem1
StepHypRef Expression
1 ee4anv 2057 . . . 4  |-  ( E. y E. z E. w E. v ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ (
y  .+  z ) ]  .~  )  /\  (
( A  =  [
w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ ( w  .+  v ) ]  .~  ) )  <->  ( E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) ) )
2 an4 800 . . . . . . 7  |-  ( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ (
y  .+  z ) ]  .~  )  /\  (
( A  =  [
w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ ( w  .+  v ) ]  .~  ) )  <->  ( (
( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  /\  (
x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  ) ) )
3 eleq1 2313 . . . . . . . . . . . . 13  |-  ( A  =  [ y ]  .~  ->  ( A  e.  ( S /.  .~  ) 
<->  [ y ]  .~  e.  ( S /.  .~  ) ) )
4 eleq1 2313 . . . . . . . . . . . . 13  |-  ( B  =  [ z ]  .~  ->  ( B  e.  ( S /.  .~  ) 
<->  [ z ]  .~  e.  ( S /.  .~  ) ) )
53, 4bi2anan9 848 . . . . . . . . . . . 12  |-  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  ->  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) ) )
65adantr 453 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  (
( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <-> 
( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
) ) )
76biimpac 474 . . . . . . . . . 10  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) )
8 eqtr2 2271 . . . . . . . . . . . . 13  |-  ( ( A  =  [ y ]  .~  /\  A  =  [ w ]  .~  )  ->  [ y ]  .~  =  [ w ]  .~  )
9 eqtr2 2271 . . . . . . . . . . . . 13  |-  ( ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  )  ->  [ z ]  .~  =  [ v ]  .~  )
108, 9anim12i 551 . . . . . . . . . . . 12  |-  ( ( ( A  =  [
y ]  .~  /\  A  =  [ w ]  .~  )  /\  ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1110an4s 802 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1211adantl 454 . . . . . . . . . 10  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
13 th3qlem1.1 . . . . . . . . . . . 12  |-  .~  Er  S
1413a1i 12 . . . . . . . . . . 11  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  .~  Er  S )
15 simprl 735 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  =  [
w ]  .~  )
16 erdm 6556 . . . . . . . . . . . . . . . 16  |-  (  .~  Er  S  ->  dom  .~  =  S )
1713, 16ax-mp 10 . . . . . . . . . . . . . . 15  |-  dom  .~  =  S
18 simpll 733 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  e.  ( S /.  .~  )
)
19 ecelqsdm 6615 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ y ]  .~  e.  ( S /.  .~  ) )  ->  y  e.  S
)
2017, 18, 19sylancr 647 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  e.  S )
2114, 20erth 6590 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .~  w  <->  [ y ]  .~  =  [ w ]  .~  ) )
2215, 21mpbird 225 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  .~  w )
23 simprr 736 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  =  [
v ]  .~  )
24 simplr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  e.  ( S /.  .~  )
)
25 ecelqsdm 6615 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ z ]  .~  e.  ( S /.  .~  ) )  ->  z  e.  S
)
2617, 24, 25sylancr 647 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  e.  S )
2714, 26erth 6590 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
z  .~  v  <->  [ z ]  .~  =  [ v ]  .~  ) )
2823, 27mpbird 225 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  .~  v )
2915, 18eqeltrrd 2328 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ w ]  .~  e.  ( S /.  .~  ) )
30 ecelqsdm 6615 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ w ]  .~  e.  ( S /.  .~  ) )  ->  w  e.  S )
3117, 29, 30sylancr 647 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  w  e.  S )
3223, 24eqeltrrd 2328 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ v ]  .~  e.  ( S /.  .~  )
)
33 ecelqsdm 6615 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ v ]  .~  e.  ( S /.  .~  ) )  ->  v  e.  S
)
3417, 32, 33sylancr 647 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  v  e.  S )
35 th3qlem1.3 . . . . . . . . . . . . 13  |-  ( ( ( y  e.  S  /\  w  e.  S
)  /\  ( z  e.  S  /\  v  e.  S ) )  -> 
( ( y  .~  w  /\  z  .~  v
)  ->  ( y  .+  z )  .~  (
w  .+  v )
) )
3620, 31, 26, 34, 35syl22anc 1188 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
( y  .~  w  /\  z  .~  v
)  ->  ( y  .+  z )  .~  (
w  .+  v )
) )
3722, 28, 36mp2and 663 . . . . . . . . . . 11  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .+  z )  .~  ( w  .+  v
) )
3814, 37erthi 6592 . . . . . . . . . 10  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ ( y  .+  z ) ]  .~  =  [
( w  .+  v
) ]  .~  )
397, 12, 38syl2anc 645 . . . . . . . . 9  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  [ (
y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
40 eqeq12 2265 . . . . . . . . 9  |-  ( ( x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  ( x  =  u  <->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  ) )
4139, 40syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( (
x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  x  =  u ) )
4241expimpd 589 . . . . . . 7  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  /\  ( x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  ) )  ->  x  =  u ) )
432, 42syl5bi 210 . . . . . 6  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  ( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
4443exlimdvv 2026 . . . . 5  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( E. w E. v ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  ( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
4544exlimdvv 2026 . . . 4  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( E. y E. z E. w E. v ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  ( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
461, 45syl5bir 211 . . 3  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
4746alrimivv 2013 . 2  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  ->  A. x A. u ( ( E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
48 eqeq1 2259 . . . . . 6  |-  ( x  =  u  ->  (
x  =  [ ( y  .+  z ) ]  .~  <->  u  =  [ ( y  .+  z ) ]  .~  ) )
4948anbi2d 687 . . . . 5  |-  ( x  =  u  ->  (
( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ (
y  .+  z ) ]  .~  )  <->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )
) )
50492exbidv 2007 . . . 4  |-  ( x  =  u  ->  ( E. y E. z ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ ( y  .+  z ) ]  .~  ) 
<->  E. y E. z
( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [ (
y  .+  z ) ]  .~  ) ) )
51 eceq1 6582 . . . . . . . 8  |-  ( y  =  w  ->  [ y ]  .~  =  [
w ]  .~  )
5251eqeq2d 2264 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  [ y ]  .~  <->  A  =  [
w ]  .~  )
)
53 eceq1 6582 . . . . . . . 8  |-  ( z  =  v  ->  [ z ]  .~  =  [
v ]  .~  )
5453eqeq2d 2264 . . . . . . 7  |-  ( z  =  v  ->  ( B  =  [ z ]  .~  <->  B  =  [
v ]  .~  )
)
5552, 54bi2anan9 848 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  <->  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )
56 oveq12 5719 . . . . . . . 8  |-  ( ( y  =  w  /\  z  =  v )  ->  ( y  .+  z
)  =  ( w 
.+  v ) )
57 eceq1 6582 . . . . . . . 8  |-  ( ( y  .+  z )  =  ( w  .+  v )  ->  [ ( y  .+  z ) ]  .~  =  [
( w  .+  v
) ]  .~  )
5856, 57syl 17 . . . . . . 7  |-  ( ( y  =  w  /\  z  =  v )  ->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
5958eqeq2d 2264 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( u  =  [
( y  .+  z
) ]  .~  <->  u  =  [ ( w  .+  v ) ]  .~  ) )
6055, 59anbi12d 694 . . . . 5  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )  <->  ( ( A  =  [
w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ ( w  .+  v ) ]  .~  ) ) )
6160cbvex2v 2050 . . . 4  |-  ( E. y E. z ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [ ( y  .+  z ) ]  .~  ) 
<->  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )
6250, 61syl6bb 254 . . 3  |-  ( x  =  u  ->  ( E. y E. z ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ ( y  .+  z ) ]  .~  ) 
<->  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) ) )
6362mo4 2146 . 2  |-  ( E* x E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  <->  A. x A. u ( ( E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
6447, 63sylibr 205 1  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  ->  E* x E. y E. z ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   E*wmo 2115   class class class wbr 3920   dom cdm 4580  (class class class)co 5710    Er wer 6543   [cec 6544   /.cqs 6545
This theorem is referenced by:  th3qlem2  6651
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-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713  df-er 6546  df-ec 6548  df-qs 6552
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