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Theorem reuxfr 4648
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhyp 4650 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 467 . . 3  |-  ( ( T.  /\  y  e.  B )  ->  A  e.  B )
3 reuxfr.2 . . . 4  |-  ( x  e.  B  ->  E! y  e.  B  x  =  A )
43adantl 467 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  E! y  e.  B  x  =  A )
5 reuxfr.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
62, 4, 5reuxfrd 4647 . 2  |-  ( T. 
->  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
)
76trud 1446 1  |-  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437   T. wtru 1438    e. wcel 1870   E!wreu 2784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-v 3089
This theorem is referenced by:  zmax  11261  rebtwnz  11263
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