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Theorem euexex 2030
Description: Existential uniqueness and "at most one" double quantification. (Contributed by Jim Kingdon, 28-Dec-2018.)
Hypothesis
Ref Expression
euexex.1  |-  F/ y
ph
Assertion
Ref Expression
euexex  |-  ( ( E! x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )

Proof of Theorem euexex
StepHypRef Expression
1 eu5 1992 . . 3  |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
2 nfmo1 1957 . . . . . 6  |-  F/ x E* x ph
3 nfa1 1477 . . . . . . 7  |-  F/ x A. x E* y ps
4 nfe1 1428 . . . . . . . 8  |-  F/ x E. x ( ph  /\  ps )
54nfmo 1965 . . . . . . 7  |-  F/ x E* y E. x (
ph  /\  ps )
63, 5nfim 1507 . . . . . 6  |-  F/ x
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) )
72, 6nfim 1507 . . . . 5  |-  F/ x
( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
8 euexex.1 . . . . . . 7  |-  F/ y
ph
98nfmo 1965 . . . . . . 7  |-  F/ y E* x ph
10 mopick 2023 . . . . . . . . 9  |-  ( ( E* x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
1110ex 113 . . . . . . . 8  |-  ( E* x ph  ->  ( E. x ( ph  /\  ps )  ->  ( ph  ->  ps ) ) )
1211com3r 78 . . . . . . 7  |-  ( ph  ->  ( E* x ph  ->  ( E. x (
ph  /\  ps )  ->  ps ) ) )
138, 9, 12alrimd 1544 . . . . . 6  |-  ( ph  ->  ( E* x ph  ->  A. y ( E. x ( ph  /\  ps )  ->  ps )
) )
14 moim 2009 . . . . . . 7  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
1514spsd 1474 . . . . . 6  |-  ( A. y ( E. x
( ph  /\  ps )  ->  ps )  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
1613, 15syl6 33 . . . . 5  |-  ( ph  ->  ( E* x ph  ->  ( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) ) )
177, 16exlimi 1528 . . . 4  |-  ( E. x ph  ->  ( E* x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) ) )
1817imp 122 . . 3  |-  ( ( E. x ph  /\  E* x ph )  -> 
( A. x E* y ps  ->  E* y E. x ( ph  /\ 
ps ) ) )
191, 18sylbi 119 . 2  |-  ( E! x ph  ->  ( A. x E* y ps 
->  E* y E. x
( ph  /\  ps )
) )
2019imp 122 1  |-  ( ( E! x ph  /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   F/wnf 1392   E.wex 1424   E!weu 1945   E*wmo 1946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by:  mosubt  2783  funco  5021
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