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Theorem moexex 2356
Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
Hypothesis
Ref Expression
moexex.1  |-  F/ y
ph
Assertion
Ref Expression
moexex  |-  ( ( E* x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )

Proof of Theorem moexex
StepHypRef Expression
1 nfmo1 2274 . . . 4  |-  F/ x E* x ph
2 nfa1 1833 . . . . 5  |-  F/ x A. x E* y ps
3 nfe1 1780 . . . . . 6  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2280 . . . . 5  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 1855 . . . 4  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
6 moexex.1 . . . . . . 7  |-  F/ y
ph
76nfmo 2280 . . . . . 6  |-  F/ y E* x ph
8 mopick 2346 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
98ex 434 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
109com23 78 . . . . . 6  |-  ( E* x ph  ->  ( ph  ->  ( E. x
( ph  /\  ps )  ->  ps ) ) )
117, 6, 10alrimd 1817 . . . . 5  |-  ( E* x ph  ->  ( ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
12 moim 2326 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1312spsd 1804 . . . . 5  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1411, 13syl6 33 . . . 4  |-  ( E* x ph  ->  ( ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
151, 5, 14exlimd 1849 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
166nfex 1883 . . . . . . 7  |-  F/ y E. x ph
17 exsimpl 1645 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1816, 17exlimi 1847 . . . . . 6  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
19 exmo 2289 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2019ori 375 . . . . . 6  |-  ( -. 
E. y E. x
( ph  /\  ps )  ->  E* y E. x
( ph  /\  ps )
)
2118, 20nsyl4 142 . . . . 5  |-  ( -. 
E* y E. x
( ph  /\  ps )  ->  E. x ph )
2221con1i 129 . . . 4  |-  ( -. 
E. x ph  ->  E* y E. x (
ph  /\  ps )
)
2322a1d 25 . . 3  |-  ( -. 
E. x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) )
2415, 23pm2.61d1 159 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2524imp 429 1  |-  ( ( E* x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587   F/wnf 1590   E*wmo 2261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-eu 2264  df-mo 2265
This theorem is referenced by:  moexexv  2358  2moswap  2363
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