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Theorem sb8e 1829
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
sb8e.1  |-  F/ y
ph
Assertion
Ref Expression
sb8e  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )

Proof of Theorem sb8e
StepHypRef Expression
1 sb8e.1 . 2  |-  F/ y
ph
21nfs1 1781 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1744 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvex 1729 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   F/wnf 1436   E.wex 1468   [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sb8mo  2011  mo2n  2025  mor  2039  nfrexdya  2468
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