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Theorem sb8 2032
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 1766 . . 3  |-  F/ y A. x ph
3 stdpc4 1964 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
42, 3alrimi 1745 . 2  |-  ( A. x ph  ->  A. y [ y  /  x ] ph )
51nfs1 1984 . . 3  |-  F/ x [ y  /  x ] ph
6 stdpc7 1858 . . 3  |-  ( y  =  x  ->  ( [ y  /  x ] ph  ->  ph ) )
75, 1, 6cbv3 1922 . 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 1527   F/wnf 1531   [wsb 1629
This theorem is referenced by:  sb8e  2033  sbhb  2046  sbnf2  2047  sb8eu  2161  sb8iota  5226  mo5f  23143
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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