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Theorem cbveu 1967
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 1956 . 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 1716 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
65eubii 1952 . 2  |-  ( E! y [ y  /  x ] ph  <->  E! y ps )
72, 6bitri 182 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1390   [wsb 1687   E!weu 1943
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946
This theorem is referenced by:  cbvmo  1983  cbvreu  2580  cbvreucsf  2975  tz6.12f  5254  f1ompt  5372  climeu  10336
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