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Theorem sbalv 2165
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 2163 . 2  |-  ( [ y  /  x ] A. z ph  <->  A. z [ y  /  x ] ph )
2 sbalv.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32albii 1572 . 2  |-  ( A. z [ y  /  x ] ph  <->  A. z ps )
41, 3bitri 241 1  |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wal 1546   [wsb 1655
This theorem is referenced by:  sbmo  2270  sbabel  2551  mo5f  23818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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