MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  moexexOLD Structured version   Unicode version

Theorem moexexOLD 2359
Description: Obsolete proof of moexex 2358 as of 28-Dec.2018. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
moexex.1  |-  F/ y
ph
Assertion
Ref Expression
moexexOLD  |-  ( ( E* x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )

Proof of Theorem moexexOLD
StepHypRef Expression
1 nfmo1 2276 . . . . 5  |-  F/ x E* x ph
2 nfa1 1836 . . . . . 6  |-  F/ x A. x E* y ps
3 nfe1 1780 . . . . . . 7  |-  F/ x E. x ( ph  /\  ps )
43nfmo 2282 . . . . . 6  |-  F/ x E* y E. x (
ph  /\  ps )
52, 4nfim 1858 . . . . 5  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
61, 5nfim 1858 . . . 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 2282 . . . . . 6  |-  F/ y E* x ph
9 mopick 2348 . . . . . . . 8  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
109ex 434 . . . . . . 7  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1110com3r 79 . . . . . 6  |-  ( ph  ->  ( E* x ph  ->  ( E. x (
ph  /\  ps )  ->  ps ) ) )
127, 8, 11alrimd 1820 . . . . 5  |-  ( ph  ->  ( E* x ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
13 moim 2328 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1413spsd 1807 . . . . 5  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1512, 14syl6 33 . . . 4  |-  ( ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
166, 15exlimi 1850 . . 3  |-  ( E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
177nfex 1886 . . . . . . . 8  |-  F/ y E. x ph
18 exsimpl 1645 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  E. x ph )
1917, 18exlimi 1850 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  ->  E. x ph )
2019con3i 135 . . . . . 6  |-  ( -. 
E. x ph  ->  -. 
E. y E. x
( ph  /\  ps )
)
21 exmo 2291 . . . . . . 7  |-  ( E. y E. x (
ph  /\  ps )  \/  E* y E. x
( ph  /\  ps )
)
2221ori 375 . . . . . 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 25 . . . 4  |-  ( -. 
E. x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) )
2524a1d 25 . . 3  |-  ( -. 
E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\  ps ) ) ) )
2616, 25pm2.61i 164 . 2  |-  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2726imp 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 2263
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 1955
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 2266  df-mo 2267
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator