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Theorem a9e2ndALT 29023
Description: If at least two sets exist (dtru 4217) , then the same is true expressed in an alternate form similar to the form of a9e 1904. 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 29000. (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 2804 . . . . . . 7  |-  u  e. 
_V
2 a9e 1904 . . . . . . 7  |-  E. y 
y  =  v
31, 2pm3.2i 441 . . . . . 6  |-  ( u  e.  _V  /\  E. y  y  =  v
)
4 19.42v 1858 . . . . . . 7  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  ( u  e.  _V  /\  E. y 
y  =  v ) )
54biimpri 197 . . . . . 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 2805 . . . . . . 7  |-  ( u  e.  _V  <->  E. x  x  =  u )
87anbi1i 676 . . . . . 6  |-  ( ( u  e.  _V  /\  y  =  v )  <->  ( E. x  x  =  u  /\  y  =  v ) )
98exbii 1572 . . . . 5  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  E. y
( E. x  x  =  u  /\  y  =  v ) )
106, 9mpbi 199 . . . 4  |-  E. y
( E. x  x  =  u  /\  y  =  v )
11 id 19 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  y )
12 hbnae 1908 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. y  -.  A. x  x  =  y )
13 hbn1 1716 . . . . . . . . . . . 12  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
14 ax-17 1606 . . . . . . . . . . . . . . . 16  |-  ( z  =  v  ->  A. x  z  =  v )
15 ax-17 1606 . . . . . . . . . . . . . . . 16  |-  ( y  =  v  ->  A. z 
y  =  v )
16 id 19 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  z  =  y )
17 equequ1 1667 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
z  =  v  <->  y  =  v ) )
1817a1i 10 . . . . . . . . . . . . . . . . 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 1917 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2111, 20syl 15 . . . . . . . . . . . . . 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 1549 . . . . . . . . . . . 12  |-  ( A. x  -.  A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2413, 23syl 15 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2511, 24syl 15 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
26 19.41rg 28615 . . . . . . . . . 10  |-  ( A. x ( y  =  v  ->  A. x  y  =  v )  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
2725, 26syl 15 . . . . . . . . 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 1549 . . . . . . 7  |-  ( A. y  -.  A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3012, 29syl 15 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3111, 30syl 15 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
32 exim 1565 . . . . 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 15 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( E. y ( E. x  x  =  u  /\  y  =  v )  ->  E. y E. x
( x  =  u  /\  y  =  v ) ) )
34 pm3.35 570 . . . 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 644 . . 3  |-  ( -. 
A. x  x  =  y  ->  E. y E. x ( x  =  u  /\  y  =  v ) )
36 excomim 1797 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  v )  ->  E. x E. y
( x  =  u  /\  y  =  v ) )
3735, 36syl 15 . 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 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
  Copyright terms: Public domain W3C validator