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Theorem sb6rf 1825
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 1741 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
32equcoms 1684 . . . 4  |-  ( y  =  x  ->  ( ph  ->  [ y  /  x ] ph ) )
43com12 30 . . 3  |-  ( ph  ->  ( y  =  x  ->  [ y  /  x ] ph ) )
51, 4alrimih 1445 . 2  |-  ( ph  ->  A. y ( y  =  x  ->  [ y  /  x ] ph ) )
6 sb2 1740 . . 3  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  [ x  /  y ] [ y  /  x ] ph )
71sbid2h 1821 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
86, 7sylib 121 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  ph )
95, 8impbii 125 1  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329   [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-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-sb 1736
This theorem is referenced by:  2sb6rf  1963  eu1  2022
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