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Theorem moxfr 26733
Description: Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.)
Hypotheses
Ref Expression
moxfr.a  |-  A  e. 
_V
moxfr.b  |-  E! y  x  =  A
moxfr.c  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
moxfr  |-  ( E* x ph  <->  E* y ps )
Distinct variable groups:    ps, x    ph, y    x, A    x, y
Allowed substitution hints:    ph( x)    ps( y)    A( y)

Proof of Theorem moxfr
StepHypRef Expression
1 moxfr.a . . . . . 6  |-  A  e. 
_V
21a1i 11 . . . . 5  |-  ( y  e.  _V  ->  A  e.  _V )
3 moxfr.b . . . . . . . 8  |-  E! y  x  =  A
4 euex 2304 . . . . . . . 8  |-  ( E! y  x  =  A  ->  E. y  x  =  A )
53, 4ax-mp 8 . . . . . . 7  |-  E. y  x  =  A
6 rexv 2970 . . . . . . 7  |-  ( E. y  e.  _V  x  =  A  <->  E. y  x  =  A )
75, 6mpbir 201 . . . . . 6  |-  E. y  e.  _V  x  =  A
87a1i 11 . . . . 5  |-  ( x  e.  _V  ->  E. y  e.  _V  x  =  A )
9 moxfr.c . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
102, 8, 9rexxfr 4743 . . . 4  |-  ( E. x  e.  _V  ph  <->  E. y  e.  _V  ps )
11 rexv 2970 . . . 4  |-  ( E. x  e.  _V  ph  <->  E. x ph )
12 rexv 2970 . . . 4  |-  ( E. y  e.  _V  ps  <->  E. y ps )
1310, 11, 123bitr3i 267 . . 3  |-  ( E. x ph  <->  E. y ps )
141, 3, 9euxfr 3120 . . 3  |-  ( E! x ph  <->  E! y ps )
1513, 14imbi12i 317 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
16 df-mo 2286 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
17 df-mo 2286 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
1815, 16, 173bitr4i 269 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2281   E*wmo 2282   E.wrex 2706   _Vcvv 2956
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-v 2958
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