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Theorem reuxfr2 4747
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
reuxfr2.1  |-  ( y  e.  B  ->  A  e.  B )
reuxfr2.2  |-  ( x  e.  B  ->  E* y  e.  B x  =  A )
Assertion
Ref Expression
reuxfr2  |-  ( E! x  e.  B  E. y  e.  B  (
x  =  A  /\  ph )  <->  E! y  e.  B  ph )
Distinct variable groups:    ph, x    x, A    x, y, B
Allowed substitution hints:    ph( y)    A( y)

Proof of Theorem reuxfr2
StepHypRef Expression
1 reuxfr2.1 . . . 4  |-  ( y  e.  B  ->  A  e.  B )
21adantl 453 . . 3  |-  ( (  T.  /\  y  e.  B )  ->  A  e.  B )
3 reuxfr2.2 . . . 4  |-  ( x  e.  B  ->  E* y  e.  B x  =  A )
43adantl 453 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  E* y  e.  B x  =  A )
52, 4reuxfr2d 4746 . 2  |-  (  T. 
->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ph )  <->  E! y  e.  B  ph ) )
65trud 1332 1  |-  ( E! x  e.  B  E. y  e.  B  (
x  =  A  /\  ph )  <->  E! y  e.  B  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    T. wtru 1325    = wceq 1652    e. wcel 1725   E.wrex 2706   E!wreu 2707   E*wrmo 2708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-v 2958
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