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Theorem sb5rf 1780
Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb5rf  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . . 4  |-  ( ph  ->  A. y ph )
21sbid2h 1777 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
3 sb1 1696 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  ->  E. y
( y  =  x  /\  [ y  /  x ] ph ) )
42, 3sylbir 133 . 2  |-  ( ph  ->  E. y ( y  =  x  /\  [
y  /  x ] ph ) )
5 stdpc7 1700 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] ph  ->  ph ) )
65imp 122 . . 3  |-  ( ( y  =  x  /\  [ y  /  x ] ph )  ->  ph )
71, 6exlimih 1529 . 2  |-  ( E. y ( y  =  x  /\  [ y  /  x ] ph )  ->  ph )
84, 7impbii 124 1  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426   [wsb 1692
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-sb 1693
This theorem is referenced by:  2sb5rf  1913  sbelx  1921
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