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Theorem 2reurmo 31945
Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2376. (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 31941 . 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 3084 . . 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 2830 1  |-  ( E! x  e.  A  E* y  e.  B  ph  ->  E* x  e.  A  E! y  e.  B  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   E!wreu 2819   E*wrmo 2820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-eu 2279  df-mo 2280  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825
This theorem is referenced by: (None)
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