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Theorem moexex 2390
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 2330 . . . 4  |-  F/ x E* x ph
2 nfa1 1999 . . . . 5  |-  F/ x A. x E* y ps
3 nfe1 1935 . . . . . 6  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2336 . . . . 5  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 2023 . . . 4  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
6 moexex.1 . . . . . . 7  |-  F/ y
ph
76nfmo 2336 . . . . . 6  |-  F/ y E* x ph
8 mopick 2384 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
98ex 441 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
109com23 80 . . . . . 6  |-  ( E* x ph  ->  ( ph  ->  ( E. x
( ph  /\  ps )  ->  ps ) ) )
117, 6, 10alrimd 1979 . . . . 5  |-  ( E* x ph  ->  ( ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
12 moim 2368 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1312spsd 1965 . . . . 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 2017 . . 3  |-  ( E* x ph  ->  ( E. x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
166nfex 2050 . . . . . . 7  |-  F/ y E. x ph
17 exsimpl 1737 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1816, 17exlimi 2015 . . . . . 6  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
19 exmo 2344 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2019ori 382 . . . . . 6  |-  ( -. 
E. y E. x
( ph  /\  ps )  ->  E* y E. x
( ph  /\  ps )
)
2118, 20nsyl4 149 . . . . 5  |-  ( -. 
E* y E. x
( ph  /\  ps )  ->  E. x ph )
2221con1i 134 . . . 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 164 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2524imp 436 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 376   A.wal 1450   E.wex 1671   F/wnf 1675   E*wmo 2320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324
This theorem is referenced by:  moexexv  2391  2moswap  2396
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