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Theorem moexex 2212
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 2154 . . . . 5  |-  F/ x E* x ph
2 nfa1 1756 . . . . . 6  |-  F/ x A. x E* y ps
3 nfe1 1706 . . . . . . 7  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2160 . . . . . 6  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 1769 . . . . 5  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
61, 5nfim 1769 . . . 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 2160 . . . . . 6  |-  F/ y E* x ph
9 mopick 2205 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
109ex 423 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1110com3r 73 . . . . . 6  |-  ( ph  ->  ( E* x ph  ->  ( E. x (
ph  /\  ps )  ->  ps ) ) )
127, 8, 11alrimd 1749 . . . . 5  |-  ( ph  ->  ( E* x ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
13 moim 2189 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1413spsd 1740 . . . . 5  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1512, 14syl6 29 . . . 4  |-  ( ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
166, 15exlimi 1801 . . 3  |-  ( E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
177nfex 1767 . . . . . . . 8  |-  F/ y E. x ph
18 exsimpl 1579 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1917, 18exlimi 1801 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
2019con3i 127 . . . . . 6  |-  ( -. 
E. x ph  ->  -. 
E. y E. x
( ph  /\  ps )
)
21 exmo 2188 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2221ori 364 . . . . . 6  |-  ( -. 
E. y E. x
( ph  /\  ps )  ->  E* y E. x
( ph  /\  ps )
)
2320, 22syl 15 . . . . 5  |-  ( -. 
E. x ph  ->  E* y E. x (
ph  /\  ps )
)
2423a1d 22 . . . 4  |-  ( -. 
E. x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) )
2524a1d 22 . . 3  |-  ( -. 
E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) ) )
2616, 25pm2.61i 156 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2726imp 418 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 358   A.wal 1527   E.wex 1528   F/wnf 1531   E*wmo 2144
This theorem is referenced by:  moexexv  2213  2moswap  2218
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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