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Theorem reuxfr2 4558
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 4557 . 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 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-v 2790
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