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

Theorem sbalv 2184
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 2181 . 2  |-  ( [ y  /  x ] A. z ph  <->  A. z [ y  /  x ] ph )
2 sbalv.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32albii 1611 . 2  |-  ( A. z [ y  /  x ] ph  <->  A. z ps )
41, 3bitri 249 1  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wal 1368   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  sbmo  2322  sbabel  2643  mo5f  26013
  Copyright terms: Public domain W3C validator