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Theorem 2rexreu 27939
Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2358. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2rexreu  |-  ( ( E! x  e.  A  E. y  e.  B  ph 
/\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E! y  e.  B  ph )
Distinct variable groups:    y, A    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x)    B( y)

Proof of Theorem 2rexreu
StepHypRef Expression
1 reurmo 2923 . . . 4  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A E. y  e.  B  ph )
2 reurex 2922 . . . . 5  |-  ( E! y  e.  B  ph  ->  E. y  e.  B  ph )
32rmoimi 27930 . . . 4  |-  ( E* x  e.  A E. y  e.  B  ph  ->  E* x  e.  A E! y  e.  B  ph )
41, 3syl 16 . . 3  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A E! y  e.  B  ph )
5 2reurex 27935 . . 3  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  E. x  e.  A  E! y  e.  B  ph )
64, 5anim12ci 551 . 2  |-  ( ( E! x  e.  A  E. y  e.  B  ph 
/\  E! y  e.  B  E. x  e.  A  ph )  -> 
( E. x  e.  A  E! y  e.  B  ph  /\  E* x  e.  A E! y  e.  B  ph )
)
7 reu5 2921 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  E! y  e.  B  ph  /\  E* x  e.  A E! y  e.  B  ph ) )
86, 7sylibr 204 1  |-  ( ( E! x  e.  A  E. y  e.  B  ph 
/\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E! y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wrex 2706   E!wreu 2707   E*wrmo 2708
This theorem is referenced by:  2reu1  27940  2reu2  27941  2reu3  27942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713
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