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Theorem sb8eh 1777
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 1730 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1695 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvexh 1679 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  exsb  1926
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