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Theorem sb8 1752
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 1706 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1670 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 1653 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 102   A.wal 1257   F/wnf 1365   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662
This theorem is referenced by:  sbnf2  1873  sb8eu  1929  nfraldya  2375  rabeq0  3275  abeq0  3276  sb8iota  4902
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