MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbalv Unicode version

Theorem sbalv 2081
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 2079 . 2  |-  ( [ y  /  x ] A. z ph  <->  A. z [ y  /  x ] ph )
2 sbalv.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32albii 1556 . 2  |-  ( A. z [ y  /  x ] ph  <->  A. z ps )
41, 3bitri 240 1  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wal 1530   [wsb 1638
This theorem is referenced by:  sbmo  2186  sbabel  2458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
  Copyright terms: Public domain W3C validator