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Theorem sb8 2045
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb8  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb8
StepHypRef Expression
1 sb5rf.1 . . . 4  |-  F/ y
ph
21nfal 1778 . . 3  |-  F/ y A. x ph
3 stdpc4 1977 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
42, 3alrimi 1757 . 2  |-  ( A. x ph  ->  A. y [ y  /  x ] ph )
51nfs1 1997 . . 3  |-  F/ x [ y  /  x ] ph
6 stdpc7 1870 . . 3  |-  ( y  =  x  ->  ( [ y  /  x ] ph  ->  ph ) )
75, 1, 6cbv3 1935 . 2  |-  ( A. y [ y  /  x ] ph  ->  A. x ph )
84, 7impbii 180 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530   F/wnf 1534   [wsb 1638
This theorem is referenced by:  sb8e  2046  sbhb  2059  sbnf2  2060  sb8eu  2174  sb8iota  5242  mo5f  23159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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