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Theorem th3qlem1 6658
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
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 ee4anv 1946 . . . 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 586 . . . . . . 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 2252 . . . . . . . . . . . . 13  |-  ( A  =  [ y ]  .~  ->  ( A  e.  ( S /.  .~  ) 
<->  [ y ]  .~  e.  ( S /.  .~  ) ) )
4 eleq1 2252 . . . . . . . . . . . . 13  |-  ( B  =  [ z ]  .~  ->  ( B  e.  ( S /.  .~  ) 
<->  [ z ]  .~  e.  ( S /.  .~  ) ) )
53, 4bi2anan9 606 . . . . . . . . . . . 12  |-  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  ->  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) ) )
65adantr 276 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  (
( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <-> 
( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
) ) )
76biimpac 298 . . . . . . . . . 10  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) )
8 eqtr2 2208 . . . . . . . . . . . . 13  |-  ( ( A  =  [ y ]  .~  /\  A  =  [ w ]  .~  )  ->  [ y ]  .~  =  [ w ]  .~  )
9 eqtr2 2208 . . . . . . . . . . . . 13  |-  ( ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  )  ->  [ z ]  .~  =  [ v ]  .~  )
108, 9anim12i 338 . . . . . . . . . . . 12  |-  ( ( ( A  =  [
y ]  .~  /\  A  =  [ w ]  .~  )  /\  ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1110an4s 588 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1211adantl 277 . . . . . . . . . 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 9 . . . . . . . . . . 11  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  .~  Er  S )
15 simprl 529 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  =  [
w ]  .~  )
16 erdm 6564 . . . . . . . . . . . . . . . 16  |-  (  .~  Er  S  ->  dom  .~  =  S )
1713, 16ax-mp 5 . . . . . . . . . . . . . . 15  |-  dom  .~  =  S
18 simpll 527 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  e.  ( S /.  .~  )
)
19 ecelqsdm 6626 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ y ]  .~  e.  ( S /.  .~  ) )  ->  y  e.  S
)
2017, 18, 19sylancr 414 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  e.  S )
2114, 20erth 6600 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .~  w  <->  [ y ]  .~  =  [ w ]  .~  ) )
2215, 21mpbird 167 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  .~  w )
23 simprr 531 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  =  [
v ]  .~  )
24 simplr 528 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  e.  ( S /.  .~  )
)
25 ecelqsdm 6626 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ z ]  .~  e.  ( S /.  .~  ) )  ->  z  e.  S
)
2617, 24, 25sylancr 414 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  e.  S )
2714, 26erth 6600 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
z  .~  v  <->  [ z ]  .~  =  [ v ]  .~  ) )
2823, 27mpbird 167 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  .~  v )
2915, 18eqeltrrd 2267 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ w ]  .~  e.  ( S /.  .~  ) )
30 ecelqsdm 6626 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ w ]  .~  e.  ( S /.  .~  ) )  ->  w  e.  S )
3117, 29, 30sylancr 414 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  w  e.  S )
3223, 24eqeltrrd 2267 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ v ]  .~  e.  ( S /.  .~  )
)
33 ecelqsdm 6626 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ v ]  .~  e.  ( S /.  .~  ) )  ->  v  e.  S
)
3417, 32, 33sylancr 414 . . . . . . . . . . . . 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 1250 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
( y  .~  w  /\  z  .~  v
)  ->  ( y  .+  z )  .~  (
w  .+  v )
) )
3722, 28, 36mp2and 433 . . . . . . . . . . 11  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .+  z )  .~  ( w  .+  v
) )
3814, 37erthi 6602 . . . . . . . . . 10  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ ( y  .+  z ) ]  .~  =  [
( w  .+  v
) ]  .~  )
397, 12, 38syl2anc 411 . . . . . . . . 9  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  [ (
y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
40 eqeq12 2202 . . . . . . . . 9  |-  ( ( x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  ( x  =  u  <->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  ) )
4139, 40syl5ibrcom 157 . . . . . . . 8  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( (
x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  x  =  u ) )
4241expimpd 363 . . . . . . 7  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  /\  ( x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  ) )  ->  x  =  u ) )
432, 42biimtrid 152 . . . . . 6  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  ( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
4443exlimdvv 1909 . . . . 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 1909 . . . 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, 45biimtrrid 153 . . 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 1886 . 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 2196 . . . . . 6  |-  ( x  =  u  ->  (
x  =  [ ( y  .+  z ) ]  .~  <->  u  =  [ ( y  .+  z ) ]  .~  ) )
4948anbi2d 464 . . . . 5  |-  ( x  =  u  ->  (
( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ (
y  .+  z ) ]  .~  )  <->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )
) )
50492exbidv 1879 . . . 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 6589 . . . . . . . 8  |-  ( y  =  w  ->  [ y ]  .~  =  [
w ]  .~  )
5251eqeq2d 2201 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  [ y ]  .~  <->  A  =  [
w ]  .~  )
)
53 eceq1 6589 . . . . . . . 8  |-  ( z  =  v  ->  [ z ]  .~  =  [
v ]  .~  )
5453eqeq2d 2201 . . . . . . 7  |-  ( z  =  v  ->  ( B  =  [ z ]  .~  <->  B  =  [
v ]  .~  )
)
5552, 54bi2anan9 606 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  <->  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )
56 oveq12 5901 . . . . . . . 8  |-  ( ( y  =  w  /\  z  =  v )  ->  ( y  .+  z
)  =  ( w 
.+  v ) )
5756eceq1d 6590 . . . . . . 7  |-  ( ( y  =  w  /\  z  =  v )  ->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
5857eqeq2d 2201 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( u  =  [
( y  .+  z
) ]  .~  <->  u  =  [ ( w  .+  v ) ]  .~  ) )
5955, 58anbi12d 473 . . . . 5  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )  <->  ( ( A  =  [
w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ ( w  .+  v ) ]  .~  ) ) )
6059cbvex2v 1936 . . . 4  |-  ( E. y E. z ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [ ( y  .+  z ) ]  .~  ) 
<->  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )
6150, 60bitrdi 196 . . 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 ) ]  .~  ) ) )
6261mo4 2099 . 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 )
)
6347, 62sylibr 134 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 4    /\ wa 104    <-> wb 105   A.wal 1362    = wceq 1364   E.wex 1503   E*wmo 2039    e. wcel 2160   class class class wbr 4018   dom cdm 4641  (class class class)co 5892    Er wer 6551   [cec 6552   /.cqs 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fv 5240  df-ov 5895  df-er 6554  df-ec 6556  df-qs 6560
This theorem is referenced by:  th3qlem2  6659
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