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Theorem sb8e 2169
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb8e  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )

Proof of Theorem sb8e
StepHypRef Expression
1 sb5rf.1 . 2  |-  F/ y
ph
21nfs1 2105 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1993 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvex 2023 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   E.wex 1613   F/wnf 1617   [wsb 1740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741
This theorem is referenced by:  sbnf2  2184  2sb8e  2212  sb8mo  2321  mo3  2324  mo3OLD  2325  mopickOLD  2357  sbcexf  30766  exlimddvfi  30775  pm11.58  31543  bnj985  34197
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