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Theorem sb6rf 2031
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb6rf  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)

Proof of Theorem sb6rf
StepHypRef Expression
1 sb5rf.1 . . 3  |-  F/ y
ph
2 sbequ1 1859 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
32equcoms 1651 . . . 4  |-  ( y  =  x  ->  ( ph  ->  [ y  /  x ] ph ) )
43com12 27 . . 3  |-  ( ph  ->  ( y  =  x  ->  [ y  /  x ] ph ) )
51, 4alrimi 1745 . 2  |-  ( ph  ->  A. y ( y  =  x  ->  [ y  /  x ] ph ) )
6 sb2 1963 . . 3  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  [ x  /  y ] [ y  /  x ] ph )
71sbid2 2024 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
86, 7sylib 188 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  ph )
95, 8impbii 180 1  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   F/wnf 1531   [wsb 1629
This theorem is referenced by:  2sb6rf  2057  eu1  2164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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