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Theorem a9e2ndeq 28325
Description: "At least two sets exist" expressed in the form of dtru 4201 is logically equivalent to the same expressed in a form similar to a9e 1891 if dtru 4201 is false implies  u  =  v. a9e2ndeq 28325 is derived from a9e2ndeqVD 28685. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
a9e2ndeq  |-  ( ( -.  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 a9e2ndeq
StepHypRef Expression
1 a9e2nd 28324 . . 3  |-  ( -. 
A. x  x  =  y  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
2 a9e2eq 28323 . . . 4  |-  ( A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y
( x  =  u  /\  y  =  v ) ) )
31a1d 22 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v ) ) )
42, 3pm2.61i 156 . . 3  |-  ( u  =  v  ->  E. x E. y ( x  =  u  /\  y  =  v ) )
51, 4jaoi 368 . 2  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  ->  E. x E. y ( x  =  u  /\  y  =  v )
)
6 olc 373 . . . 4  |-  ( u  =  v  ->  ( -.  A. x  x  =  y  \/  u  =  v ) )
76a1d 22 . . 3  |-  ( u  =  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
) )
8 excom 1786 . . . . . 6  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  <->  E. y E. x ( x  =  u  /\  y  =  v )
)
9 neeq1 2454 . . . . . . . . . . . . 13  |-  ( x  =  u  ->  (
x  =/=  v  <->  u  =/=  v ) )
109biimprcd 216 . . . . . . . . . . . 12  |-  ( u  =/=  v  ->  (
x  =  u  ->  x  =/=  v ) )
1110adantrd 454 . . . . . . . . . . 11  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  x  =/=  v ) )
12 simpr 447 . . . . . . . . . . . 12  |-  ( ( x  =  u  /\  y  =  v )  ->  y  =  v )
1312a1i 10 . . . . . . . . . . 11  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  y  =  v ) )
14 neeq2 2455 . . . . . . . . . . . 12  |-  ( y  =  v  ->  (
x  =/=  y  <->  x  =/=  v ) )
1514biimprcd 216 . . . . . . . . . . 11  |-  ( x  =/=  v  ->  (
y  =  v  ->  x  =/=  y ) )
1611, 13, 15ee22 1352 . . . . . . . . . 10  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  x  =/=  y ) )
17 sp 1716 . . . . . . . . . . 11  |-  ( A. x  x  =  y  ->  x  =  y )
1817necon3ai 2486 . . . . . . . . . 10  |-  ( x  =/=  y  ->  -.  A. x  x  =  y )
1916, 18syl6 29 . . . . . . . . 9  |-  ( u  =/=  v  ->  (
( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y )
)
2019eximdv 1608 . . . . . . . 8  |-  ( u  =/=  v  ->  ( E. x ( x  =  u  /\  y  =  v )  ->  E. x  -.  A. x  x  =  y ) )
21 nfnae 1896 . . . . . . . . 9  |-  F/ x  -.  A. x  x  =  y
222119.9 1783 . . . . . . . 8  |-  ( E. x  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
2320, 22syl6ib 217 . . . . . . 7  |-  ( u  =/=  v  ->  ( E. x ( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y ) )
2423eximdv 1608 . . . . . 6  |-  ( u  =/=  v  ->  ( E. y E. x ( x  =  u  /\  y  =  v )  ->  E. y  -.  A. x  x  =  y
) )
258, 24syl5bi 208 . . . . 5  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  E. y  -.  A. x  x  =  y
) )
26 nfnae 1896 . . . . . 6  |-  F/ y  -.  A. x  x  =  y
272619.9 1783 . . . . 5  |-  ( E. y  -.  A. x  x  =  y  <->  -.  A. x  x  =  y )
2825, 27syl6ib 217 . . . 4  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  -.  A. x  x  =  y ) )
29 orc 374 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. x  x  =  y  \/  u  =  v ) )
3028, 29syl6 29 . . 3  |-  ( u  =/=  v  ->  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
) )
317, 30pm2.61ine 2522 . 2  |-  ( E. x E. y ( x  =  u  /\  y  =  v )  ->  ( -.  A. x  x  =  y  \/  u  =  v )
)
325, 31impbii 180 1  |-  ( ( -.  A. x  x  =  y  \/  u  =  v )  <->  E. x E. y ( x  =  u  /\  y  =  v ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    =/= wne 2446
This theorem is referenced by:  2sb5nd  28326  2uasbanh  28327  2sb5ndVD  28686  2uasbanhVD  28687  2sb5ndALT  28709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ne 2448  df-v 2790
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