ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sb8h Unicode version

Theorem sb8h 1865
Description: Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb8h.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
sb8h  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb8h
StepHypRef Expression
1 sb8h.1 . 2  |-  ( ph  ->  A. y ph )
21hbsb3 1819 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1782 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvalh 1764 1  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1362   [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by:  sbhb  1952  sb8euh  2061
  Copyright terms: Public domain W3C validator