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Theorem reuxfr 4673
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhyp 4675 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
Hypotheses
Ref Expression
reuxfr.1  |-  ( y  e.  B  ->  A  e.  B )
reuxfr.2  |-  ( x  e.  B  ->  E! y  e.  B  x  =  A )
reuxfr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
reuxfr  |-  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
Distinct variable groups:    ps, x    ph, y    x, A    x, y, B
Allowed substitution hints:    ph( x)    ps( y)    A( y)

Proof of Theorem reuxfr
StepHypRef Expression
1 reuxfr.1 . . . 4  |-  ( y  e.  B  ->  A  e.  B )
21adantl 466 . . 3  |-  ( ( T.  /\  y  e.  B )  ->  A  e.  B )
3 reuxfr.2 . . . 4  |-  ( x  e.  B  ->  E! y  e.  B  x  =  A )
43adantl 466 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  E! y  e.  B  x  =  A )
5 reuxfr.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
62, 4, 5reuxfrd 4672 . 2  |-  ( T. 
->  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
)
76trud 1388 1  |-  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   T. wtru 1380    e. wcel 1767   E!wreu 2816
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-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-v 3115
This theorem is referenced by:  zmax  11179  rebtwnz  11181
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