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Theorem cbvmo 2300
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 2075 . . 3  |-  ( E. x ph  <->  E. y ps )
51, 2, 3cbveu 2299 . . 3  |-  ( E! x ph  <->  E! y ps )
64, 5imbi12i 327 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
7 df-mo 2268 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
8 df-mo 2268 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
96, 7, 83bitr4i 280 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   E.wex 1659   F/wnf 1663   E!weu 2263   E*wmo 2264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268
This theorem is referenced by:  dffun6f  5607  opabiotafun  5934  2ndcdisj  20448  cbvdisjf  28162  phpreu  31837
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