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Theorem cbvmo 1988
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 1686 . . 3  |-  ( E. x ph  <->  E. y ps )
51, 2, 3cbveu 1972 . . 3  |-  ( E! x ph  <->  E! y ps )
64, 5imbi12i 237 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
7 df-mo 1952 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
8 df-mo 1952 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
96, 7, 83bitr4i 210 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394   E.wex 1426   E!weu 1948   E*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  dffun6f  5015
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