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Theorem ax6e2ndALT 36966
Description: If at least two sets exist (dtru 4616) , then the same is true expressed in an alternate form similar to the form of ax6e 2058. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndVD 36944. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2ndALT  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
Distinct variable groups:    x, u    y, u    x, v

Proof of Theorem ax6e2ndALT
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 3090 . . . . . . 7  |-  u  e. 
_V
2 ax6e 2058 . . . . . . 7  |-  E. y 
y  =  v
31, 2pm3.2i 456 . . . . . 6  |-  ( u  e.  _V  /\  E. y  y  =  v
)
4 19.42v 1826 . . . . . . 7  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  ( u  e.  _V  /\  E. y 
y  =  v ) )
54biimpri 209 . . . . . 6  |-  ( ( u  e.  _V  /\  E. y  y  =  v )  ->  E. y
( u  e.  _V  /\  y  =  v ) )
63, 5ax-mp 5 . . . . 5  |-  E. y
( u  e.  _V  /\  y  =  v )
7 isset 3091 . . . . . . 7  |-  ( u  e.  _V  <->  E. x  x  =  u )
87anbi1i 699 . . . . . 6  |-  ( ( u  e.  _V  /\  y  =  v )  <->  ( E. x  x  =  u  /\  y  =  v ) )
98exbii 1714 . . . . 5  |-  ( E. y ( u  e. 
_V  /\  y  =  v )  <->  E. y
( E. x  x  =  u  /\  y  =  v ) )
106, 9mpbi 211 . . . 4  |-  E. y
( E. x  x  =  u  /\  y  =  v )
11 id 23 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  -.  A. x  x  =  y )
12 hbnae 2113 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  A. y  -.  A. x  x  =  y )
13 hbn1 1890 . . . . . . . . . . . 12  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
14 ax-5 1751 . . . . . . . . . . . . . . . 16  |-  ( z  =  v  ->  A. x  z  =  v )
15 ax-5 1751 . . . . . . . . . . . . . . . 16  |-  ( y  =  v  ->  A. z 
y  =  v )
16 id 23 . . . . . . . . . . . . . . . . 17  |-  ( z  =  y  ->  z  =  y )
17 equequ1 1850 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  y  ->  (
z  =  v  <->  y  =  v ) )
1817a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( z  =  y  -> 
z  =  y )  ->  ( z  =  y  ->  ( z  =  v  <->  y  =  v ) ) )
1916, 18ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( z  =  y  ->  (
z  =  v  <->  y  =  v ) )
2014, 15, 19dvelimh 2134 . . . . . . . . . . . . . . 15  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2111, 20syl 17 . . . . . . . . . . . . . 14  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2221idiALT 36468 . . . . . . . . . . . . 13  |-  ( -. 
A. x  x  =  y  ->  ( y  =  v  ->  A. x  y  =  v )
)
2322alimi 1680 . . . . . . . . . . . 12  |-  ( A. x  -.  A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2413, 23syl 17 . . . . . . . . . . 11  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
2511, 24syl 17 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  A. x
( y  =  v  ->  A. x  y  =  v ) )
26 19.41rg 36553 . . . . . . . . . 10  |-  ( A. x ( y  =  v  ->  A. x  y  =  v )  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
2725, 26syl 17 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x
( x  =  u  /\  y  =  v ) ) )
2827idiALT 36468 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  ( ( E. x  x  =  u  /\  y  =  v )  ->  E. x
( x  =  u  /\  y  =  v ) ) )
2928alimi 1680 . . . . . . 7  |-  ( A. y  -.  A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3012, 29syl 17 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
3111, 30syl 17 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  A. y
( ( E. x  x  =  u  /\  y  =  v )  ->  E. x ( x  =  u  /\  y  =  v ) ) )
32 exim 1701 . . . . 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 17 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( E. y ( E. x  x  =  u  /\  y  =  v )  ->  E. y E. x
( x  =  u  /\  y  =  v ) ) )
34 pm3.35 589 . . . 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 667 . . 3  |-  ( -. 
A. x  x  =  y  ->  E. y E. x ( x  =  u  /\  y  =  v ) )
36 excomim 1903 . . 3  |-  ( E. y E. x ( x  =  u  /\  y  =  v )  ->  E. x E. y
( x  =  u  /\  y  =  v ) )
3735, 36syl 17 . 2  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
3837idiALT 36468 1  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   _Vcvv 3087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator