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Theorem sb8 2151
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 2088 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1976 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 2005 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1379   F/wnf 1601   [wsb 1724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1598  df-nf 1602  df-sb 1725
This theorem is referenced by:  sbhb  2166  sbnf2  2167  sb8eu  2302  sb8euOLD  2303  sb8iota  5544  mo5f  27248  wl-sb8eut  29990  sbcalf  30485  ax11-pm2  34121  bj-nfcf  34204
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