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Theorem 2reu2rex 27630
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2313. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2reu2rex  |-  ( E! x  e.  A  E! y  e.  B  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 2reu2rex
StepHypRef Expression
1 reurex 2866 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  ->  E. x  e.  A  E! y  e.  B  ph )
2 reurex 2866 . . 3  |-  ( E! y  e.  B  ph  ->  E. y  e.  B  ph )
32reximi 2757 . 2  |-  ( E. x  e.  A  E! y  e.  B  ph  ->  E. x  e.  A  E. y  e.  B  ph )
41, 3syl 16 1  |-  ( E! x  e.  A  E! y  e.  B  ph  ->  E. x  e.  A  E. y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wrex 2651   E!wreu 2652
This theorem is referenced by:  2reu1  27633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658
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