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Theorem sb6rf 1781
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
sb5rf.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb6rf  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3  |-  ( ph  ->  A. y ph )
2 sbequ1 1698 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
32equcoms 1641 . . . 4  |-  ( y  =  x  ->  ( ph  ->  [ y  /  x ] ph ) )
43com12 30 . . 3  |-  ( ph  ->  ( y  =  x  ->  [ y  /  x ] ph ) )
51, 4alrimih 1403 . 2  |-  ( ph  ->  A. y ( y  =  x  ->  [ y  /  x ] ph ) )
6 sb2 1697 . . 3  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  [ x  /  y ] [ y  /  x ] ph )
71sbid2h 1777 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
86, 7sylib 120 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  ph )
95, 8impbii 124 1  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287   [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:  2sb6rf  1914  eu1  1973
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