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Theorem 2eu1 2348
Description: Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
Assertion
Ref Expression
2eu1  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )

Proof of Theorem 2eu1
StepHypRef Expression
1 2eu2ex 2340 . . . . 5  |-  ( E! x E! y ph  ->  E. x E. y ph )
2 df-mo 2268 . . . . . . 7  |-  ( E* y ph  <->  ( E. y ph  ->  E! y ph ) )
32albii 1687 . . . . . 6  |-  ( A. x E* y ph  <->  A. x
( E. y ph  ->  E! y ph )
)
4 euim 2317 . . . . . . 7  |-  ( ( E. x E. y ph  /\  A. x ( E. y ph  ->  E! y ph ) )  ->  ( E! x E! y ph  ->  E! x E. y ph )
)
54ex 435 . . . . . 6  |-  ( E. x E. y ph  ->  ( A. x ( E. y ph  ->  E! y ph )  -> 
( E! x E! y ph  ->  E! x E. y ph )
) )
63, 5syl5bi 220 . . . . 5  |-  ( E. x E. y ph  ->  ( A. x E* y ph  ->  ( E! x E! y ph  ->  E! x E. y ph ) ) )
71, 6syl 17 . . . 4  |-  ( E! x E! y ph  ->  ( A. x E* y ph  ->  ( E! x E! y ph  ->  E! x E. y ph ) ) )
87pm2.43b 52 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  E! x E. y ph ) )
9 2euswap 2342 . . . 4  |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
108, 9syld 45 . . 3  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  E! y E. x ph ) )
118, 10jcad 535 . 2  |-  ( A. x E* y ph  ->  ( E! x E! y
ph  ->  ( E! x E. y ph  /\  E! y E. x ph )
) )
12 2exeu 2343 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
1311, 12impbid1 206 1  |-  ( A. x E* y ph  ->  ( E! x E! y
ph 
<->  ( E! x E. y ph  /\  E! y E. x ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1659   E!weu 2263   E*wmo 2264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-eu 2267  df-mo 2268
This theorem is referenced by:  2eu2  2350  2eu3  2351  2eu5  2353
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