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Theorem sb8 2218
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 2157 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 2047 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 2075 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wal 1435   F/wnf 1663   [wsb 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by:  sbhb  2233  sbnf2  2234  sb8eu  2299  sb8iota  5569  mo5f  28106  ax11-pm2  31396  bj-nfcf  31484  wl-sb8eut  31820  sbcalf  32266
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