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

Theorem sbalv 1924
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbalv.1  |-  ( [ y  /  x ] ph 
<->  ps )
Assertion
Ref Expression
sbalv  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Distinct variable groups:    x, z    y,
z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)

Proof of Theorem sbalv
StepHypRef Expression
1 sbal 1919 . 2  |-  ( [ y  /  x ] A. z ph  <->  A. z [ y  /  x ] ph )
2 sbalv.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32albii 1400 . 2  |-  ( A. z [ y  /  x ] ph  <->  A. z ps )
41, 3bitri 182 1  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283   [wsb 1687
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbmo  2002  sbabel  2248  peano2  4372
  Copyright terms: Public domain W3C validator