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Theorem 2eu4 2405
Description: This theorem provides us with a definition of double existential uniqueness ("exactly one 
x and exactly one  y"). Naively one might think (incorrectly) that it could be defined by  E! x E! y ph. See 2eu1 2402 for a condition under which the naive definition holds and 2exeu 2398 for a one-way implication. See 2eu5 2406 and 2eu8 2409 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
Assertion
Ref Expression
2eu4  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Distinct variable groups:    x, y,
z, w    ph, z, w
Allowed substitution hints:    ph( x, y)

Proof of Theorem 2eu4
StepHypRef Expression
1 eu5 2345 . . 3  |-  ( E! x E. y ph  <->  ( E. x E. y ph  /\  E* x E. y ph ) )
2 eu5 2345 . . . 4  |-  ( E! y E. x ph  <->  ( E. y E. x ph  /\  E* y E. x ph ) )
3 excom 1944 . . . . 5  |-  ( E. y E. x ph  <->  E. x E. y ph )
43anbi1i 709 . . . 4  |-  ( ( E. y E. x ph  /\  E* y E. x ph )  <->  ( E. x E. y ph  /\  E* y E. x ph ) )
52, 4bitri 257 . . 3  |-  ( E! y E. x ph  <->  ( E. x E. y ph  /\  E* y E. x ph ) )
61, 5anbi12i 711 . 2  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( ( E. x E. y ph  /\ 
E* x E. y ph )  /\  ( E. x E. y ph  /\ 
E* y E. x ph ) ) )
7 anandi 844 . 2  |-  ( ( E. x E. y ph  /\  ( E* x E. y ph  /\  E* y E. x ph )
)  <->  ( ( E. x E. y ph  /\ 
E* x E. y ph )  /\  ( E. x E. y ph  /\ 
E* y E. x ph ) ) )
8 2mo2 2399 . . 3  |-  ( ( E* x E. y ph  /\  E* y E. x ph )  <->  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w )
) )
98anbi2i 708 . 2  |-  ( ( E. x E. y ph  /\  ( E* x E. y ph  /\  E* y E. x ph )
)  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w )
) ) )
106, 7, 93bitr2i 281 1  |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   E!weu 2319   E*wmo 2320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324
This theorem is referenced by:  2eu5  2406  2eu6  2407
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