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Theorem cbveu 2163
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
Hypotheses
Ref Expression
cbveu.1  |-  F/ y
ph
cbveu.2  |-  F/ x ps
cbveu.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbveu  |-  ( E! x ph  <->  E! y ps )

Proof of Theorem cbveu
StepHypRef Expression
1 cbveu.1 . . 3  |-  F/ y
ph
21sb8eu 2161 . 2  |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
3 cbveu.2 . . . 4  |-  F/ x ps
4 cbveu.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
53, 4sbie 1978 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
65eubii 2152 . 2  |-  ( E! y [ y  /  x ] ph  <->  E! y ps )
72, 6bitri 240 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   [wsb 1629   E!weu 2143
This theorem is referenced by:  cbvmo  2180  cbvreu  2762  cbvreucsf  3145  tz6.12f  5546  f1ompt  5682  climeu  12029
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
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