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Theorem sb6rfNEW7 29529
Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1NEW  |-  F/ y
ph
Assertion
Ref Expression
sb6rfNEW7  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)

Proof of Theorem sb6rfNEW7
StepHypRef Expression
1 sb5rf.1NEW . . 3  |-  F/ y
ph
2 sbequ1 1943 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
32equcoms 1693 . . . 4  |-  ( y  =  x  ->  ( ph  ->  [ y  /  x ] ph ) )
43com12 29 . . 3  |-  ( ph  ->  ( y  =  x  ->  [ y  /  x ] ph ) )
51, 4alrimi 1781 . 2  |-  ( ph  ->  A. y ( y  =  x  ->  [ y  /  x ] ph ) )
6 sb2NEW7 29474 . . 3  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  [ x  /  y ] [ y  /  x ] ph )
71sbid2NEW7 29520 . . 3  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  ph )
86, 7sylib 189 . 2  |-  ( A. y ( y  =  x  ->  [ y  /  x ] ph )  ->  ph )
95, 8impbii 181 1  |-  ( ph  <->  A. y ( y  =  x  ->  [ y  /  x ] ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   F/wnf 1553   [wsb 1658
This theorem is referenced by:  2sb6rfOLD7  29699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-7v 29379
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
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