MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reuxfr Unicode version

Theorem reuxfr 4689
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhyp 4691 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 453 . . 3  |-  ( (  T.  /\  y  e.  B )  ->  A  e.  B )
3 reuxfr.2 . . . 4  |-  ( x  e.  B  ->  E! y  e.  B  x  =  A )
43adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  E! y  e.  B  x  =  A )
5 reuxfr.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
62, 4, 5reuxfrd 4688 . 2  |-  (  T. 
->  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
)
76trud 1329 1  |-  ( E! x  e.  B  ph  <->  E! y  e.  B  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    T. wtru 1322    = wceq 1649    e. wcel 1717   E!wreu 2651
This theorem is referenced by:  zmax  10503  rebtwnz  10505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-v 2901
  Copyright terms: Public domain W3C validator