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Theorem cbveu 2305
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 2301 . 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 2110 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
65eubii 2287 . 2  |-  ( E! y [ y  /  x ] ph  <->  E! y ps )
72, 6bitri 249 1  |-  ( E! x ph  <->  E! y ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   F/wnf 1590   [wsb 1702   E!weu 2262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266
This theorem is referenced by:  cbvmo  2306  cbvreu  3049  cbvreucsf  3428  tz6.12f  5816  f1ompt  5973  climeu  13150
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