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Theorem a9e2ndALT 29104
Description: If at least two sets exist (dtru 4392) , then the same is true expressed in an alternate form similar to the form of a9e 1953. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in a9e2ndVD 29082. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9e2ndALT  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
Distinct variable groups:    x, u    y, u    x, v

Proof of Theorem a9e2ndALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . . . . 7  |-  u  e. 
_V
2 a9e 1953 . . . . . . 7  |-  E. y 
y  =  v
31, 2pm3.2i 443 . . . . . 6  |-  ( u  e.  _V  /\  E. y  y  =  v
)
4 19.42v 1929 . . . . . . 7  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  ( u  e.  _V  /\  E. y 
y  =  v ) )
54biimpri 199 . . . . . 6  |-  ( ( u  e.  _V  /\  E. y  y  =  v )  ->  E. y
( u  e.  _V  /\  y  =  v ) )
63, 5ax-mp 8 . . . . 5  |-  E. y
( u  e.  _V  /\  y  =  v )
7 isset 2962 . . . . . . 7  |-  ( u  e.  _V  <->  E. x  x  =  u )
87anbi1i 678 . . . . . 6  |-  ( ( u  e.  _V  /\  y  =  v )  <->  ( E. x  x  =  u  /\  y  =  v ) )
98exbii 1593 . . . . 5  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  E. y
( E. x  x  =  u  /\  y  =  v ) )
106, 9mpbi 201 . . . 4  |-  E. y
( E. x  x  =  u  /\  y  =  v )
11 id 21 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  y )
12 hbnae 2044 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. y  -.  A. x  x  =  y )
13 hbn1 1746 . . . . . . . . . . . 12  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
14 ax-17 1627 . . . . . . . . . . . . . . . 16  |-  ( z  =  v  ->  A. x  z  =  v )
15 ax-17 1627 . . . . . . . . . . . . . . . 16  |-  ( y  =  v  ->  A. z 
y  =  v )
16 id 21 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  z  =  y )
17 equequ1 1697 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
z  =  v  <->  y  =  v ) )
1817a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  y  -> 
z  =  y )  ->  ( z  =  y  ->  ( z  =  v  <->  y  =  v ) ) )
1916, 18ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( z  =  y  ->  (
z  =  v  <->  y  =  v ) )
2014, 15, 19dvelimh 2072 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2111, 20syl 16 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2221idi 2 . . . . . . . . . . . . 13  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2322alimi 1569 . . . . . . . . . . . 12  |-  ( A. x  -.  A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2413, 23syl 16 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2511, 24syl 16 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
26 19.41rg 28699 . . . . . . . . . 10  |-  ( A. x ( y  =  v  ->  A. x  y  =  v )  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
2725, 26syl 16 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x
( x  =  u  /\  y  =  v ) ) )
2827idi 2 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x
( x  =  u  /\  y  =  v ) ) )
2928alimi 1569 . . . . . . 7  |-  ( A. y  -.  A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3012, 29syl 16 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3111, 30syl 16 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
32 exim 1585 . . . . 5  |-  ( A. y ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x
( x  =  u  /\  y  =  v ) )  ->  ( E. y ( E. x  x  =  u  /\  y  =  v )  ->  E. y E. x
( x  =  u  /\  y  =  v ) ) )
3331, 32syl 16 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( E. y ( E. x  x  =  u  /\  y  =  v )  ->  E. y E. x
( x  =  u  /\  y  =  v ) ) )
34 pm3.35 572 . . . 4  |-  ( ( E. y ( E. x  x  =  u  /\  y  =  v )  /\  ( E. y ( E. x  x  =  u  /\  y  =  v )  ->  E. y E. x
( x  =  u  /\  y  =  v ) ) )  ->  E. y E. x ( x  =  u  /\  y  =  v )
)
3510, 33, 34sylancr 646 . . 3  |-  ( -. 
A. x  x  =  y  ->  E. y E. x ( x  =  u  /\  y  =  v ) )
36 excomim 1758 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  v )  ->  E. x E. y
( x  =  u  /\  y  =  v ) )
3735, 36syl 16 . 2  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
3837idi 2 1  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-v 2960
  Copyright terms: Public domain W3C validator