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Theorem cbvmo 2015
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
cbvmo.1  |-  F/ y
ph
cbvmo.2  |-  F/ x ps
cbvmo.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvmo  |-  ( E* x ph  <->  E* y ps )

Proof of Theorem cbvmo
StepHypRef Expression
1 cbvmo.1 . . . 4  |-  F/ y
ph
2 cbvmo.2 . . . 4  |-  F/ x ps
3 cbvmo.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvex 1712 . . 3  |-  ( E. x ph  <->  E. y ps )
51, 2, 3cbveu 1999 . . 3  |-  ( E! x ph  <->  E! y ps )
64, 5imbi12i 238 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
7 df-mo 1979 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
8 df-mo 1979 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
96, 7, 83bitr4i 211 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1419   E.wex 1451   E!weu 1975   E*wmo 1976
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by:  dffun6f  5104
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