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Theorem sb8eh 1865
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8eh.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb8eh  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )

Proof of Theorem sb8eh
StepHypRef Expression
1 sb8eh.1 . 2  |-  ( ph  ->  A. y ph )
21hbsb3 1818 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1781 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvexh 1765 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1361   E.wex 1502   [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-sb 1773
This theorem is referenced by:  exsb  2019
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