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Theorem 2rexreu 27375
Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2220. (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 2755 . . . 4  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A E. y  e.  B  ph )
2 reurex 2754 . . . . 5  |-  ( E! y  e.  B  ph  ->  E. y  e.  B  ph )
32rmoimi 27366 . . . 4  |-  ( E* x  e.  A E. y  e.  B  ph  ->  E* x  e.  A E! y  e.  B  ph )
41, 3syl 15 . . 3  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A E! y  e.  B  ph )
5 2reurex 27371 . . 3  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  E. x  e.  A  E! y  e.  B  ph )
64, 5anim12ci 550 . 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 2753 . 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 203 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 358   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  2reu1  27376  2reu2  27377  2reu3  27378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551
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