ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8 Unicode version

Theorem sb8 1784
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 1737 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1701 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbval 1684 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1287   F/wnf 1394   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sbnf2  1905  sb8eu  1961  nfraldya  2412  rabeq0  3312  abeq0  3313  sb8iota  4987
  Copyright terms: Public domain W3C validator