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Theorem 2rexreu 38616
Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2380. (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 3012 . . . 4  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A  E. y  e.  B  ph )
2 reurex 3011 . . . . 5  |-  ( E! y  e.  B  ph  ->  E. y  e.  B  ph )
32rmoimi 38607 . . . 4  |-  ( E* x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A  E! y  e.  B  ph )
41, 3syl 17 . . 3  |-  ( E! x  e.  A  E. y  e.  B  ph  ->  E* x  e.  A  E! y  e.  B  ph )
5 2reurex 38612 . . 3  |-  ( E! y  e.  B  E. x  e.  A  ph  ->  E. x  e.  A  E! y  e.  B  ph )
64, 5anim12ci 571 . 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 3010 . 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 216 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 setvar class
Syntax hints:    -> wi 4    /\ wa 371   E.wrex 2740   E!wreu 2741   E*wrmo 2742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747
This theorem is referenced by:  2reu1  38617  2reu2  38618  2reu3  38619
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