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Theorem moexex 2349
Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.)
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 2291 . . . . 5  |-  F/ x E* x ph
2 nfa1 1806 . . . . . 6  |-  F/ x A. x E* y ps
3 nfe1 1747 . . . . . . 7  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2297 . . . . . 6  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 1832 . . . . 5  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
61, 5nfim 1832 . . . 4  |-  F/ x
( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
7 moexex.1 . . . . . 6  |-  F/ y
ph
87nfmo 2297 . . . . . 6  |-  F/ y E* x ph
9 mopick 2342 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
109ex 424 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1110com3r 75 . . . . . 6  |-  ( ph  ->  ( E* x ph  ->  ( E. x (
ph  /\  ps )  ->  ps ) ) )
127, 8, 11alrimd 1785 . . . . 5  |-  ( ph  ->  ( E* x ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
13 moim 2326 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1413spsd 1771 . . . . 5  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1512, 14syl6 31 . . . 4  |-  ( ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
166, 15exlimi 1821 . . 3  |-  ( E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
177nfex 1865 . . . . . . . 8  |-  F/ y E. x ph
18 exsimpl 1602 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1917, 18exlimi 1821 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
2019con3i 129 . . . . . 6  |-  ( -. 
E. x ph  ->  -. 
E. y E. x
( ph  /\  ps )
)
21 exmo 2325 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2221ori 365 . . . . . 6  |-  ( -. 
E. y E. x
( ph  /\  ps )  ->  E* y E. x
( ph  /\  ps )
)
2320, 22syl 16 . . . . 5  |-  ( -. 
E. x ph  ->  E* y E. x (
ph  /\  ps )
)
2423a1d 23 . . . 4  |-  ( -. 
E. x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) )
2524a1d 23 . . 3  |-  ( -. 
E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) ) )
2616, 25pm2.61i 158 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2726imp 419 1  |-  ( ( E* x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553   E*wmo 2281
This theorem is referenced by:  moexexv  2350  2moswap  2355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285
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