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Theorem rexxfr 4327
Description: Transfer existence from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfr.1  |-  ( y  e.  C  ->  A  e.  B )
ralxfr.2  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
ralxfr.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexxfr  |-  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
Distinct variable groups:    ps, x    ph, y    x, A    x, y, B   
x, C
Allowed substitution hints:    ph( x)    ps( y)    A( y)    C( y)

Proof of Theorem rexxfr
StepHypRef Expression
1 ralxfr.1 . . . 4  |-  ( y  e.  C  ->  A  e.  B )
21adantl 273 . . 3  |-  ( ( T.  /\  y  e.  C )  ->  A  e.  B )
3 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
43adantl 273 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 ralxfr.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65adantl 273 . . 3  |-  ( ( T.  /\  x  =  A )  ->  ( ph 
<->  ps ) )
72, 4, 6rexxfrd 4322 . 2  |-  ( T. 
->  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
)
87mptru 1308 1  |-  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1299   T. wtru 1300    e. wcel 1448   E.wrex 2376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643
This theorem is referenced by: (None)
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