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Theorem 2reurmo 27927
Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2353. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
Assertion
Ref Expression
2reurmo  |-  ( 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 2reurmo
StepHypRef Expression
1 reuimrmo 27923 . 2  |-  ( A. x  e.  A  ( E! y  e.  B  ph 
->  E* y  e.  B ph )  ->  ( E! x  e.  A  E* y  e.  B ph  ->  E* x  e.  A E! y  e.  B  ph ) )
2 reurmo 2915 . . 3  |-  ( E! y  e.  B  ph  ->  E* y  e.  B ph )
32a1i 11 . 2  |-  ( x  e.  A  ->  ( E! y  e.  B  ph 
->  E* y  e.  B ph ) )
41, 3mprg 2767 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. wcel 1725   E!wreu 2699   E*wrmo 2700
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
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 2284  df-mo 2285  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705
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