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Theorem moexex 2342
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 2282 . . . 4  |-  F/ x E* x ph
2 nfa1 1956 . . . . 5  |-  F/ x A. x E* y ps
3 nfe1 1894 . . . . . 6  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2288 . . . . 5  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 1980 . . . 4  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
6 moexex.1 . . . . . . 7  |-  F/ y
ph
76nfmo 2288 . . . . . 6  |-  F/ y E* x ph
8 mopick 2336 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
98ex 435 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
109com23 81 . . . . . 6  |-  ( E* x ph  ->  ( ph  ->  ( E. x
( ph  /\  ps )  ->  ps ) ) )
117, 6, 10alrimd 1936 . . . . 5  |-  ( E* x ph  ->  ( ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
12 moim 2320 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1312spsd 1922 . . . . 5  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1411, 13syl6 34 . . . 4  |-  ( E* x ph  ->  ( ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
151, 5, 14exlimd 1974 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
166nfex 2008 . . . . . . 7  |-  F/ y E. x ph
17 exsimpl 1723 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1816, 17exlimi 1972 . . . . . 6  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
19 exmo 2296 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2019ori 376 . . . . . 6  |-  ( -. 
E. y E. x
( ph  /\  ps )  ->  E* y E. x
( ph  /\  ps )
)
2118, 20nsyl4 147 . . . . 5  |-  ( -. 
E* y E. x
( ph  /\  ps )  ->  E. x ph )
2221con1i 132 . . . 4  |-  ( -. 
E. x ph  ->  E* y E. x (
ph  /\  ps )
)
2322a1d 26 . . 3  |-  ( -. 
E. x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) )
2415, 23pm2.61d1 162 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2524imp 430 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 370   A.wal 1435   E.wex 1657   F/wnf 1661   E*wmo 2272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-eu 2275  df-mo 2276
This theorem is referenced by:  moexexv  2343  2moswap  2348
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