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Theorem sb8 2253
Description: Substitution of variable in universal quantifier. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
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 . 2  |-  F/ y
ph
21nfs1 2194 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 2083 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 2114 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wal 1442   F/wnf 1667   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by:  sbhb  2267  sbnf2  2268  sb8eu  2332  sb8iota  5553  mo5f  28120  ax11-pm2  31438  bj-nfcf  31527  wl-sb8eut  31906  sbcalf  32352
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