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Theorem rexxfr 4389
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 275 . . 3  |-  ( ( T.  /\  y  e.  C )  ->  A  e.  B )
3 ralxfr.2 . . . 4  |-  ( x  e.  B  ->  E. y  e.  C  x  =  A )
43adantl 275 . . 3  |-  ( ( T.  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 ralxfr.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
65adantl 275 . . 3  |-  ( ( T.  /\  x  =  A )  ->  ( ph 
<->  ps ) )
72, 4, 6rexxfrd 4384 . 2  |-  ( T. 
->  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
)
87mptru 1340 1  |-  ( E. x  e.  B  ph  <->  E. y  e.  C  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   T. wtru 1332    e. wcel 1480   E.wrex 2417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688
This theorem is referenced by: (None)
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