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Theorem cbvmo 2078
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 1767 . . 3  |-  ( E. x ph  <->  E. y ps )
51, 2, 3cbveu 2062 . . 3  |-  ( E! x ph  <->  E! y ps )
64, 5imbi12i 239 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
7 df-mo 2042 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
8 df-mo 2042 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
96, 7, 83bitr4i 212 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1471   E.wex 1503   E!weu 2038   E*wmo 2039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042
This theorem is referenced by:  dffun6f  5245
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