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Theorem reuxfr2 4595
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 452 . . 3  |-  ( (  T.  /\  y  e.  B )  ->  A  e.  B )
3 reuxfr2.2 . . . 4  |-  ( x  e.  B  ->  E* y  e.  B x  =  A )
43adantl 452 . . 3  |-  ( (  T.  /\  x  e.  B )  ->  E* y  e.  B x  =  A )
52, 4reuxfr2d 4594 . 2  |-  (  T. 
->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ph )  <->  E! y  e.  B  ph ) )
65trud 1314 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 176    /\ wa 358    T. wtru 1307    = wceq 1633    e. wcel 1701   E.wrex 2578   E!wreu 2579   E*wrmo 2580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-v 2824
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