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Theorem cbveu 2043
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 2032 . 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 1784 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
65eubii 2028 . 2  |-  ( E! y [ y  /  x ] ph  <->  E! y ps )
72, 6bitri 183 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1453   [wsb 1755   E!weu 2019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022
This theorem is referenced by:  cbvmo  2059  cbvreu  2694  cbvreucsf  3113  tz6.12f  5525  f1ompt  5647  climeu  11259
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