Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2reurmo Unicode version

Theorem 2reurmo 28063
Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2229. (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 28059 . 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 2768 . . 3  |-  ( E! y  e.  B  ph  ->  E* y  e.  B ph )
32a1i 10 . 2  |-  ( x  e.  A  ->  ( E! y  e.  B  ph 
->  E* y  e.  B ph ) )
41, 3mprg 2625 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 1696   E!wreu 2558   E*wrmo 2559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564
  Copyright terms: Public domain W3C validator