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Theorem sb8 1810
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8e.1  |-  F/ y
ph
Assertion
Ref Expression
sb8  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb8
StepHypRef Expression
1 sb8e.1 . 2  |-  F/ y
ph
21nfs1 1763 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1727 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 1710 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1312   F/wnf 1419   [wsb 1718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719
This theorem is referenced by:  sbnf2  1932  sb8eu  1988  nfraldya  2444  rabeq0  3360  abeq0  3361  sb8iota  5063
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