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Theorem sb5rf 2117
Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb5rf  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )

Proof of Theorem sb5rf
StepHypRef Expression
1 sb5rf.1 . . . 4  |-  F/ y
ph
21sbid2 2111 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
3 sb1 1658 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  ->  E. y
( y  =  x  /\  [ y  /  x ] ph ) )
42, 3sylbir 205 . 2  |-  ( ph  ->  E. y ( y  =  x  /\  [
y  /  x ] ph ) )
5 stdpc7 1931 . . . 4  |-  ( y  =  x  ->  ( [ y  /  x ] ph  ->  ph ) )
65imp 419 . . 3  |-  ( ( y  =  x  /\  [ y  /  x ] ph )  ->  ph )
71, 6exlimi 1811 . 2  |-  ( E. y ( y  =  x  /\  [ y  /  x ] ph )  ->  ph )
84, 7impbii 181 1  |-  ( ph  <->  E. y ( y  =  x  /\  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547   F/wnf 1550   [wsb 1655
This theorem is referenced by:  2sb5rf  2145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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