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Theorem cbvmo 2180
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 1925 . . 3  |-  ( E. x ph  <->  E. y ps )
51, 2, 3cbveu 2163 . . 3  |-  ( E! x ph  <->  E! y ps )
64, 5imbi12i 316 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. y ps  ->  E! y ps ) )
7 df-mo 2148 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
8 df-mo 2148 . 2  |-  ( E* y ps  <->  ( E. y ps  ->  E! y ps ) )
96, 7, 83bitr4i 268 1  |-  ( E* x ph  <->  E* y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528   F/wnf 1531   E!weu 2143   E*wmo 2144
This theorem is referenced by:  dffun6f  5269  opabiotafun  6291  2ndcdisj  17182  cbvdisjf  23350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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