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Theorem sb8eh 1855
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 1808 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1771 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvexh 1755 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351   E.wex 1492   [wsb 1762
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  exsb  2008
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