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Theorem ax6e2ndeq 32412
Description: "At least two sets exist" expressed in the form of dtru 4638 is logically equivalent to the same expressed in a form similar to ax6e 1971 if dtru 4638 is false implies  u  =  v. ax6e2ndeq 32412 is derived from ax6e2ndeqVD 32789. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2ndeq  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
Distinct variable groups:    x, u    y, u    x, v    y,
v

Proof of Theorem ax6e2ndeq
StepHypRef Expression
1 ax6e2nd 32411 . . 3  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
2 ax6e2eq 32410 . . . 4  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
31a1d 25 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) )
42, 3pm2.61i 164 . . 3  |-  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
51, 4jaoi 379 . 2  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  ->  E. x E. y ( x  =  u  /\  y  =  v )
)
6 olc 384 . . . 4  |-  ( u  =  v  ->  ( -.  A. x  x  =  y  \/  u  =  v ) )
76a1d 25 . . 3  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
) )
8 excom 1798 . . . . . 6  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  <->  E. y E. x ( x  =  u  /\  y  =  v )
)
9 neeq1 2748 . . . . . . . . . . . . 13  |-  ( x  =  u  ->  (
x  =/=  v  <->  u  =/=  v ) )
109biimprcd 225 . . . . . . . . . . . 12  |-  ( u  =/=  v  ->  (
x  =  u  ->  x  =/=  v ) )
1110adantrd 468 . . . . . . . . . . 11  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  x  =/=  v ) )
12 simpr 461 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  v )  ->  y  =  v )
1312a1i 11 . . . . . . . . . . 11  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  y  =  v ) )
14 neeq2 2750 . . . . . . . . . . . 12  |-  ( y  =  v  ->  (
x  =/=  y  <->  x  =/=  v ) )
1514biimprcd 225 . . . . . . . . . . 11  |-  ( x  =/=  v  ->  (
y  =  v  ->  x  =/=  y ) )
1611, 13, 15syl6c 64 . . . . . . . . . 10  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  x  =/=  y ) )
17 sp 1808 . . . . . . . . . . 11  |-  ( A. x  x  =  y  ->  x  =  y )
1817necon3ai 2695 . . . . . . . . . 10  |-  ( x  =/=  y  ->  -.  A. x  x  =  y )
1916, 18syl6 33 . . . . . . . . 9  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y )
)
2019eximdv 1686 . . . . . . . 8  |-  ( u  =/=  v  ->  ( E. x ( x  =  u  /\  y  =  v )  ->  E. x  -.  A. x  x  =  y ) )
21 nfnae 2031 . . . . . . . . 9  |-  F/ x  -.  A. x  x  =  y
222119.9 1841 . . . . . . . 8  |-  ( E. x  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
2320, 22syl6ib 226 . . . . . . 7  |-  ( u  =/=  v  ->  ( E. x ( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y ) )
2423eximdv 1686 . . . . . 6  |-  ( u  =/=  v  ->  ( E. y E. x ( x  =  u  /\  y  =  v )  ->  E. y  -.  A. x  x  =  y
) )
258, 24syl5bi 217 . . . . 5  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  E. y  -.  A. x  x  =  y
) )
26 nfnae 2031 . . . . . 6  |-  F/ y  -.  A. x  x  =  y
272619.9 1841 . . . . 5  |-  ( E. y  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
2825, 27syl6ib 226 . . . 4  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y ) )
29 orc 385 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  y  \/  u  =  v ) )
3028, 29syl6 33 . . 3  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
) )
317, 30pm2.61ine 2780 . 2  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
)
325, 31impbii 188 1  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    =/= wne 2662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-ne 2664  df-v 3115
This theorem is referenced by:  2sb5nd  32413  2uasbanh  32414  2sb5ndVD  32790  2uasbanhVD  32791  2sb5ndALT  32812
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