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Theorem 2reu2rex 27951
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2357. (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 2924 . 2  |-  ( E! x  e.  A  E! y  e.  B  ph  ->  E. x  e.  A  E! y  e.  B  ph )
2 reurex 2924 . . 3  |-  ( E! y  e.  B  ph  ->  E. y  e.  B  ph )
32reximi 2815 . 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 2708   E!wreu 2709
This theorem is referenced by:  2reu1  27954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715
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