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Theorem th3qlem1 6374
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 1857 . . . 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 553 . . . . . . 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 2150 . . . . . . . . . . . . 13  |-  ( A  =  [ y ]  .~  ->  ( A  e.  ( S /.  .~  ) 
<->  [ y ]  .~  e.  ( S /.  .~  ) ) )
4 eleq1 2150 . . . . . . . . . . . . 13  |-  ( B  =  [ z ]  .~  ->  ( B  e.  ( S /.  .~  ) 
<->  [ z ]  .~  e.  ( S /.  .~  ) ) )
53, 4bi2anan9 573 . . . . . . . . . . . 12  |-  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  ->  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) ) )
65adantr 270 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  (
( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  <-> 
( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
) ) )
76biimpac 292 . . . . . . . . . 10  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( [
y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  ) ) )
8 eqtr2 2106 . . . . . . . . . . . . 13  |-  ( ( A  =  [ y ]  .~  /\  A  =  [ w ]  .~  )  ->  [ y ]  .~  =  [ w ]  .~  )
9 eqtr2 2106 . . . . . . . . . . . . 13  |-  ( ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  )  ->  [ z ]  .~  =  [ v ]  .~  )
108, 9anim12i 331 . . . . . . . . . . . 12  |-  ( ( ( A  =  [
y ]  .~  /\  A  =  [ w ]  .~  )  /\  ( B  =  [ z ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1110an4s 555 . . . . . . . . . . 11  |-  ( ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) )  ->  ( [ y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )
1211adantl 271 . . . . . . . . . 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 498 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  =  [
w ]  .~  )
16 erdm 6282 . . . . . . . . . . . . . . . 16  |-  (  .~  Er  S  ->  dom  .~  =  S )
1713, 16ax-mp 7 . . . . . . . . . . . . . . 15  |-  dom  .~  =  S
18 simpll 496 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ y ]  .~  e.  ( S /.  .~  )
)
19 ecelqsdm 6342 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ y ]  .~  e.  ( S /.  .~  ) )  ->  y  e.  S
)
2017, 18, 19sylancr 405 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  e.  S )
2114, 20erth 6316 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .~  w  <->  [ y ]  .~  =  [ w ]  .~  ) )
2215, 21mpbird 165 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  y  .~  w )
23 simprr 499 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  =  [
v ]  .~  )
24 simplr 497 . . . . . . . . . . . . . . 15  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ z ]  .~  e.  ( S /.  .~  )
)
25 ecelqsdm 6342 . . . . . . . . . . . . . . 15  |-  ( ( dom  .~  =  S  /\  [ z ]  .~  e.  ( S /.  .~  ) )  ->  z  e.  S
)
2617, 24, 25sylancr 405 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  e.  S )
2714, 26erth 6316 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
z  .~  v  <->  [ z ]  .~  =  [ v ]  .~  ) )
2823, 27mpbird 165 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  z  .~  v )
2915, 18eqeltrrd 2165 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ w ]  .~  e.  ( S /.  .~  ) )
30 ecelqsdm 6342 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ w ]  .~  e.  ( S /.  .~  ) )  ->  w  e.  S )
3117, 29, 30sylancr 405 . . . . . . . . . . . . 13  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  w  e.  S )
3223, 24eqeltrrd 2165 . . . . . . . . . . . . . 14  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ v ]  .~  e.  ( S /.  .~  )
)
33 ecelqsdm 6342 . . . . . . . . . . . . . 14  |-  ( ( dom  .~  =  S  /\  [ v ]  .~  e.  ( S /.  .~  ) )  ->  v  e.  S
)
3417, 32, 33sylancr 405 . . . . . . . . . . . . 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 1175 . . . . . . . . . . . 12  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
( y  .~  w  /\  z  .~  v
)  ->  ( y  .+  z )  .~  (
w  .+  v )
) )
3722, 28, 36mp2and 424 . . . . . . . . . . 11  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  (
y  .+  z )  .~  ( w  .+  v
) )
3814, 37erthi 6318 . . . . . . . . . 10  |-  ( ( ( [ y ]  .~  e.  ( S /.  .~  )  /\  [ z ]  .~  e.  ( S /.  .~  )
)  /\  ( [
y ]  .~  =  [ w ]  .~  /\ 
[ z ]  .~  =  [ v ]  .~  ) )  ->  [ ( y  .+  z ) ]  .~  =  [
( w  .+  v
) ]  .~  )
397, 12, 38syl2anc 403 . . . . . . . . 9  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  [ (
y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
40 eqeq12 2100 . . . . . . . . 9  |-  ( ( x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  ( x  =  u  <->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  ) )
4139, 40syl5ibrcom 155 . . . . . . . 8  |-  ( ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  /\  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )  ->  ( (
x  =  [ ( y  .+  z ) ]  .~  /\  u  =  [ ( w  .+  v ) ]  .~  )  ->  x  =  u ) )
4241expimpd 355 . . . . . . 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 150 . . . . . 6  |-  ( ( A  e.  ( S /.  .~  )  /\  B  e.  ( S /.  .~  ) )  -> 
( ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [
( y  .+  z
) ]  .~  )  /\  ( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )  ->  x  =  u )
)
4443exlimdvv 1825 . . . . 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 1825 . . . 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 151 . . 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 1803 . 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 2094 . . . . . 6  |-  ( x  =  u  ->  (
x  =  [ ( y  .+  z ) ]  .~  <->  u  =  [ ( y  .+  z ) ]  .~  ) )
4948anbi2d 452 . . . . 5  |-  ( x  =  u  ->  (
( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  x  =  [ (
y  .+  z ) ]  .~  )  <->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )
) )
50492exbidv 1796 . . . 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 6307 . . . . . . . 8  |-  ( y  =  w  ->  [ y ]  .~  =  [
w ]  .~  )
5251eqeq2d 2099 . . . . . . 7  |-  ( y  =  w  ->  ( A  =  [ y ]  .~  <->  A  =  [
w ]  .~  )
)
53 eceq1 6307 . . . . . . . 8  |-  ( z  =  v  ->  [ z ]  .~  =  [
v ]  .~  )
5453eqeq2d 2099 . . . . . . 7  |-  ( z  =  v  ->  ( B  =  [ z ]  .~  <->  B  =  [
v ]  .~  )
)
5552, 54bi2anan9 573 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  <->  ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  ) ) )
56 oveq12 5643 . . . . . . . 8  |-  ( ( y  =  w  /\  z  =  v )  ->  ( y  .+  z
)  =  ( w 
.+  v ) )
5756eceq1d 6308 . . . . . . 7  |-  ( ( y  =  w  /\  z  =  v )  ->  [ ( y  .+  z ) ]  .~  =  [ ( w  .+  v ) ]  .~  )
5857eqeq2d 2099 . . . . . 6  |-  ( ( y  =  w  /\  z  =  v )  ->  ( u  =  [
( y  .+  z
) ]  .~  <->  u  =  [ ( w  .+  v ) ]  .~  ) )
5955, 58anbi12d 457 . . . . 5  |-  ( ( y  =  w  /\  z  =  v )  ->  ( ( ( A  =  [ y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [
( y  .+  z
) ]  .~  )  <->  ( ( A  =  [
w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ ( w  .+  v ) ]  .~  ) ) )
6059cbvex2v 1847 . . . 4  |-  ( E. y E. z ( ( A  =  [
y ]  .~  /\  B  =  [ z ]  .~  )  /\  u  =  [ ( y  .+  z ) ]  .~  ) 
<->  E. w E. v
( ( A  =  [ w ]  .~  /\  B  =  [ v ]  .~  )  /\  u  =  [ (
w  .+  v ) ]  .~  ) )
6150, 60syl6bb 194 . . 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 2009 . 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 132 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 102    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   E*wmo 1949   class class class wbr 3837   dom cdm 4428  (class class class)co 5634    Er wer 6269   [cec 6270   /.cqs 6271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fv 5010  df-ov 5637  df-er 6272  df-ec 6274  df-qs 6278
This theorem is referenced by:  th3qlem2  6375
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